When I first starting studying differential geometry, I asked my lecturer a question about smooth manifolds that didn't admit a partition of unity. He promptly told not to worry about such objects as they were only studied by the extremely eccentric. I would like to know if this is true, ie, does anyone study manifolds that don't admit a partition of unity (not whether such people are eccentric).



 Smooth manifolds that don't admit a partition of unity
The answer to your stated question ("Does anyone study non-paracompact manifolds?") is certainly yes. Here are a few papers which do just this:



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> Gauld, David.
>  Manifolds at and beyond the limit of metrisability. (English summary) Proceedings of the Kirbyfest (Berkeley, CA, 1998), 125--133 (electronic),
>  Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999. 
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> <http://www.emis.de/journals/GT/ftp/main/m2/m2-7.pdf>
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Among other things, Gauld references that there are two paracompact and two nonparacompact 1-manifolds, and $\aleph\_0$ paracompact and $2^{\aleph\_1}$ non-paracompact 2-manifolds. (That's a lot!)



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> Foliations and foliated vector bundles, 1969 MIT lecture notes
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> <http://www.foliations.org/surveys/FoliationLectNotes_Milnor.pdf>
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Milnor entertains non-paracompact manifolds. In particular he constructs a (necessarily non-paracompact) surface with uncountable fundamental group. Milnor also says: "The main object of this exercise is to imbue the reader with suitable respect for non-paracompact manifolds."



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> Balogh, Zoltan; Gruenhage, Gary.
>  Two more perfectly normal non-metrizable manifolds. (English summary)
>  Topology Appl. 151 (2005), no. 1-3, 260--272. 
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The existence of perfectly normal, non-metrizable (hence non-paracompact) manifolds is shown to depend upon one's set-theoretic assumptions. 


And so forth. I could find 10 more papers without much effort. I'm not sure I could find 100. (A MathSciNet search with "manifold" and ("nonmetrizable" or "non-paracompact") in the anywhere fields doesn't return many hits.) So some serious mathematicians take non-paracompact manifolds seriously enough to write some papers about them. On the other hand, although one could use more complimentary language than "extremely eccentric", your lecturer's take on non-paracompact manifolds seems to be an accurate reflection of how most geometric topologists feel: they seem mostly to be used as a source of counterexamples and to be of interest to general and set-theoretic topologists.


