These are also called "individual-specific intercepts", because one way to estimate the FE model is to a "least-squares dummy variables regression", in which one regresses $y$ on $x$ and a $n$ dummy variables where each individual on the panel has one dummy that takes the values one if an observation belongs to that person (household, unit, firm,...). The $\hat\alpha_i$ then estimate these intercepts, which may then be interpreted as usual intercepts in regressions, with the only difference that each intercept is specific to a single unit.
I'm just thinking out loud here,
Suppose you have industry-county-year level data, your outcome is $Y_{ict}$, and you are interested in the effect of some variable $x_{ict}$.
In your strategy you would correctly think you can use:
(1) industry-county (panel) fixed effects to control for time invariant confounding factors across these panels as well as the average difference in time varying covariates across industry-county pairs
(2) year fixed effects to control for shocks that are common to all industries and counties in a given year
However what if there are shocks that are common across some counties in regions indexed by $r$, yet are both time varying and different across regions?
That is, perhaps the true data generating process is
$Y_{ict}=\underbrace{\theta_{ic}}_\text{panel fixed effect}+\underbrace{\theta_t}_\text{year fixed effect}+\underbrace{\theta_{rt}}_\text{regional shocks}+\underbrace{\beta}_\text{parameter of interest} X_{ict}+\underbrace{\epsilon_{ict}}_\text{idiosyncratic shock}$
But you estimate a model
$Y_{ict}=\theta_{ic}+\theta_t+\beta X_{ict}+\epsilon_{ict}$
which does not attempt to proxy for this regional shock, then,to to the degree that $Cov(\theta_{rt},X_{ict})\neq 0$, I believe your estimate $\hat{\beta}$ would in part reflect the variation in $\theta_{rt}$ that covaries with $X_{ict}$.
That is,
$plim \; \hat{\beta} =\underbrace{ \beta}_\text{true parameter} + \underbrace{\frac{Cov(X_{ict},\theta_{rt})}{Var(X_{ict}}}_\text{bias}$
to solve this I believe it is possible that you could
(1) Cluster your standard errors at the geographical level where you think there may be correlated disturbances
and
2) Find an instrument $Z_{ict}$ for $X_{ict}$ that is strongly correlated with $X_{ict}$ (relevant) that has an effect on the outcome only through its effect on $X_{ict}$ and not through $\theta_{rt}$ influencing $Z_{ict}$ or through $Z_{ict}$ influencing $Y_{ict}$ directly (excludibility).
Best Answer
Including fixed effects (time, indidvidual or both) does not change the meaning of beta coefficients. You could just see fixed effects as a means to control for shocks in different time periods (i.e. time fixed effects) or individual characteristics that we cannot control for (i.e. individual fixed effects).