In my experience the answer is not really.
My favorite example of a Hausdorff manifold which is not paracompact (which incidentally is also my all-time favorite counter example to most point-set topology questions) is the Long Line.
Since it is not paracompact, it does not admit partitions of unity. All of it's homotopy groups vanish, yet it is not contractible.
If I remember correctly, its tangent bundle is non-trivial.
Even if people don't study these, then it is still important to know about the type of things which don't satisfy the usual hypotheses. You never know when you might run into one or need such a counter example.
