Found out the hard way that certain versions of Vim are not compatible with certain versions of Python. I keep my installation of Vim 7.3.712 on Dropbox so I can use it across multiple computers. While trying to run gVim on a new PC with Python 2.7.11 installed, gVim would close without warning or an error message when trying to open files.
Found out the cause was the one plugin I use that requires Python.
After a Google search, I found this StackOverflow thread that mentions newer versions of Python 2.7 may not work properly with Vim. Downgrading Python to 2.7.7 fixed the issue.