To understand what can go on, it is instructive to generate (and analyze) data that behave in the manner described.
For simplicity, let's forget about that sixth independent variable. So, the question describes regressions of one dependent variable $y$ against five independent variables $x_1, x_2, x_3, x_4, x_5$, in which
Each ordinary regression $y \sim x_i$ is significant at levels from $0.01$ to less than $0.001$.
The multiple regression $y \sim x_1 + \cdots + x_5$ yields significant coefficients only for $x_1$ and $x_2$.
All variance inflation factors (VIFs) are low, indicating good conditioning in the design matrix (that is, lack of collinearity among the $x_i$).
Let's make this happen as follows:
Generate $n$ normally distributed values for $x_1$ and $x_2$. (We will choose $n$ later.)
Let $y = x_1 + x_2 + \varepsilon$ where $\varepsilon$ is independent normal error of mean $0$. Some trial and error is needed to find a suitable standard deviation for $\varepsilon$; $1/100$ works fine (and is rather dramatic: $y$ is extremely well correlated with $x_1$ and $x_2$, even though it is only moderately correlated with $x_1$ and $x_2$ individually).
Let $x_j$ = $x_1/5 + \delta$, $j=3,4,5$, where $\delta$ is independent standard normal error. This makes $x_3,x_4,x_5$ only slightly dependent on $x_1$. However, via the tight correlation between $x_1$ and $y$, this induces a tiny correlation between $y$ and these $x_j$.
Here's the rub: if we make $n$ large enough, these slight correlations will result in significant coefficients, even though $y$ is almost entirely "explained" by only the first two variables.
I found that $n=500$ works just fine for reproducing the reported p-values. Here's a scatterplot matrix of all six variables:
By inspecting the right column (or the bottom row) you can see that $y$ has a good (positive) correlation with $x_1$ and $x_2$ but little apparent correlation with the other variables. By inspecting the rest of this matrix, you can see that the independent variables $x_1, \ldots, x_5$ appear to be mutually uncorrelated (the random $\delta$ mask the tiny dependencies we know are there.) There are no exceptional data--nothing terribly outlying or with high leverage. The histograms show that all six variables are approximately normally distributed, by the way: these data are as ordinary and "plain vanilla" as one could possibly want.
In the regression of $y$ against $x_1$ and $x_2$, the p-values are essentially 0. In the individual regressions of $y$ against $x_3$, then $y$ against $x_4$, and $y$ against $x_5$, the p-values are 0.0024, 0.0083, and 0.00064, respectively: that is, they are "highly significant." But in the full multiple regression, the corresponding p-values inflate to .46, .36, and .52, respectively: not significant at all. The reason for this is that once $y$ has been regressed against $x_1$ and $x_2$, the only stuff left to "explain" is the tiny amount of error in the residuals, which will approximate $\varepsilon$, and this error is almost completely unrelated to the remaining $x_i$. ("Almost" is correct: there is a really tiny relationship induced from the fact that the residuals were computed in part from the values of $x_1$ and $x_2$ and the $x_i$, $i=3,4,5$, do have some weak relationship to $x_1$ and $x_2$. This residual relationship is practically undetectable, though, as we saw.)
The conditioning number of the design matrix is only 2.17: that's very low, showing no indication of high multicollinearity whatsoever. (Perfect lack of collinearity would be reflected in a conditioning number of 1, but in practice this is seen only with artificial data and designed experiments. Conditioning numbers in the range 1-6 (or even higher, with more variables) are unremarkable.) This completes the simulation: it has successfully reproduced every aspect of the problem.
The important insights this analysis offers include
p-values don't tell us anything directly about collinearity. They depend strongly on the amount of data.
Relationships among p-values in multiple regressions and p-values in related regressions (involving subsets of the independent variable) are complex and usually unpredictable.
Consequently, as others have argued, p-values should not be your sole guide (or even your principal guide) to model selection.
Edit
It is not necessary for $n$ to be as large as $500$ for these phenomena to appear. Inspired by additional information in the question, the following is a dataset constructed in a similar fashion with $n=24$ (in this case $x_j = 0.4 x_1 + 0.4 x_2 + \delta$ for $j=3,4,5$). This creates correlations of 0.38 to 0.73 between $x_{1-2}$ and $x_{3-5}$. The condition number of the design matrix is 9.05: a little high, but not terrible. (Some rules of thumb say that condition numbers as high as 10 are ok.) The p-values of the individual regressions against $x_3, x_4, x_5$ are 0.002, 0.015, and 0.008: significant to highly significant. Thus, some multicollinearity is involved, but it's not so large that one would work to change it. The basic insight remains the same: significance and multicollinearity are different things; only mild mathematical constraints hold among them; and it is possible for the inclusion or exclusion of even a single variable to have profound effects on all p-values even without severe multicollinearity being an issue.
x1 x2 x3 x4 x5 y
-1.78256 -0.334959 -1.22672 -1.11643 0.233048 -2.12772
0.796957 -0.282075 1.11182 0.773499 0.954179 0.511363
0.956733 0.925203 1.65832 0.25006 -0.273526 1.89336
0.346049 0.0111112 1.57815 0.767076 1.48114 0.365872
-0.73198 -1.56574 -1.06783 -0.914841 -1.68338 -2.30272
0.221718 -0.175337 -0.0922871 1.25869 -1.05304 0.0268453
1.71033 0.0487565 -0.435238 -0.239226 1.08944 1.76248
0.936259 1.00507 1.56755 0.715845 1.50658 1.93177
-0.664651 0.531793 -0.150516 -0.577719 2.57178 -0.121927
-0.0847412 -1.14022 0.577469 0.694189 -1.02427 -1.2199
-1.30773 1.40016 -1.5949 0.506035 0.539175 0.0955259
-0.55336 1.93245 1.34462 1.15979 2.25317 1.38259
1.6934 0.192212 0.965777 0.283766 3.63855 1.86975
-0.715726 0.259011 -0.674307 0.864498 0.504759 -0.478025
-0.800315 -0.655506 0.0899015 -2.19869 -0.941662 -1.46332
-0.169604 -1.08992 -1.80457 -0.350718 0.818985 -1.2727
0.365721 1.10428 0.33128 -0.0163167 0.295945 1.48115
0.215779 2.233 0.33428 1.07424 0.815481 2.4511
1.07042 0.0490205 -0.195314 0.101451 -0.721812 1.11711
-0.478905 -0.438893 -1.54429 0.798461 -0.774219 -0.90456
1.2487 1.03267 0.958559 1.26925 1.31709 2.26846
-0.124634 -0.616711 0.334179 0.404281 0.531215 -0.747697
-1.82317 1.11467 0.407822 -0.937689 -1.90806 -0.723693
-1.34046 1.16957 0.271146 1.71505 0.910682 -0.176185
Two things to keep in mind with MR analyses: First, the VIF cut-off of 10 is to flag extreme variance inflation. You have VIF values ≈2.5, which would result in $R^2 ≈ 0.6$...this means 60% of the variance is shared amongst those variables. Thus, there is some amount of multicollinearity at play here. Second, the p-values reported by SPSS for each variable in an MR model are to assess the significance of if adding that one variable to the model with all but that variable produces a statistically significant change in the amount of variance of the predicted variable explained. (That is a very long sentence, I realize.) In other words,
• calculate $R^2$ for the entire model;
• drop the variable on the current row;
• recalculate the MR and get the smaller $R^2$;
• assess the statistical significance of $H_0: \Delta R^2=0$; and
• repeat for all of the variables.
What may be more useful to you is to use a Type-I $\Delta R^2$ calculation for the model (this requires entering the variables in the model in a fixed and specific order).
Happy to share more if it will help.
Best Answer
Including variables in your multiple regressions is something that depends on your hypothesis and what you are testing. But you can check the variance inflation factor (VIF) that is used as an indicator of multicollinearity. If VIFs are less that 10, means multicollinearity is not a problem. If VIFs for two variables is 10 or higher then you have to keep just one of those variables and eliminate the other one. Hope this help.