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This answer is based on Mercurial experience, not Git, but this experience says using Dropbox this way is asking for corrupt repositories if there's even a chance that you'll be updating the same Dropbox-based repository from different machines at various times (Mac, Unix, Windows in my case).
I don't have a complete list of the things that can go wrong, but here's a specific example that bit me. Each machine has its own notion of line-ending characters and how upper/lower case characters are handled in file names. Dropbox and Git/Mercurial handle this slightly differently (I don't recall the exact differences). If Dropbox updates the repository behind Git/Mercurial's back, presto, broken repository. This happens immediately and invisibly, so you don't even know your repository is broken until you try to recover something from it.
After digging out from one mess doing things this way, I've been using the following recipe with great success and no sign of problems. Simply move your repository out of Dropbox. Use Dropbox for everything else; documentation, JAR files, anything you please. And use GitHub (Git) or Bitbucket (Mercurial) to manage the repository itself. Both are free so this adds nothing to the costs, and each tool now plays to its strengths.
Running Git/Mercurial on top of Dropbox adds nothing except risk. Don't do it.
I've been using Mercurial in the recommended manner and urge that you be cautious, especially if any of the machines differ. The Dropbox fora are full of complaints of mysterious filename case problems turning up spontaneously. Hg (and I presume Git) won't notice or complain during routine checkins and you'll only hear about the corruption when it complains of a corrupt repo when you try to use it for real. Bad news. Wish I could be more specific about the problem and its workarounds; I'm still trying to dig out from this mess myself.
There's also an open source project (a collection of cross platform [Linux, Mac, Win] scripts) that does all the nitty-gritty details of the repository management with a handful (3-4) of commands.
https://github.com/karalabe/gitbox/wiki
Sample usage is:
$ gitbox create myapp
Creating empty repository...
Initializing new repository...
Repository successfully created.
$ gitbox clone myapp
Cloning repository...
Repository successfully cloned.
After which normal git usage:
$ echo “Some change” > somefile.txt
$ git add somefile.txt
$ git commit –m “Created some file”
$ git push
Check the project wiki and the manuals for full command reference and tutorials.