This question tells us that in general colimits do not exist in the category of manifolds.
However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its altas. In this sense, it seems that constructing manifolds via atlas is a means of completing the model category of open subsets of $R^n$ (or more precisely a comma category based on this), so it should be reasonable that at least certain kinds of colimits should hold in the category of manifolds.
Considering the altas as a "good" system, do colimits of systems which are
1) of same-dimensional manifolds
2) have only open injections as morphisms,
3) is countable
and
4) (add your additional condition here),
exist?
PS the counter example in the above referred question fails conditions 1 and 2.
EDIT: After our good discussion below, perhaps let me restate my original question. I am motivated by the fact that a manifold is a colimit of its altas. What I'm looking for is a generalization of this fact. Can we extract out some general properties of an atlas that makes its colimit exist?
For instance do colimits of countable inverse limits of open smooth injections of (neccessarily) same-dimensional manifolds exist? (remove as many quantifiers as is uncessary).
I really do mean inverse system and not direct system. Observe that the altas (if we take all finite intersections) is filtered in the direction away from the colimit.
Best Answer
A counterexample in which your first three conditions hold is the following: take two copies of the real line and glue them along the open subset $\mathbb{R}^\ast$. This can be realized as the colimit of the diagram $$ \mathbb{R}^\ast \sqcup \mathbb{R}^\ast \rightrightarrows \mathbb{R} \sqcup \mathbb{R} $$ where the first arrow is the canonical injection of the first factor into the first factor and the second into the second, and the second arrow switches the two pieces. The resulting topological spaces is the famous line with doubled origin, which is evidently non-Hausdorff.
I do not know if there is a condition you could impose to bar this example, but it does not seem probable to me that one exists.
EDIT: This is in response to Colin's comment that connectedness precludes the previous example. Take $$\mathbb{C}^\ast \rightrightarrows \mathbb{C}$$ with the first arrow being the inclusion and the second one the composition of the latter with reflection about the real line. The colimit is the closed half-plane, which is a manifold with boundary. Also, by requiring connectedness you lose coproducts!