I am using the Tor Browser Bundle, so all my internet browsing traffic is routed via Tor. However if I have other services like Dropbox, another web browser, etc. running on my machine that aren't routed through Tor, can that compromise my anonymity?
No. Or at least, there aren't supposed to be any ways for the interaction to hurt your anonymity. If you find any, please let us know!
If your non-Tor traffic is interacting with the adversary (which might be more common than you think, with centralized web resources like advertising sites), and also the adversary runs the website you're visiting over Tor, then he could potentially introduce congestion using the non-Tor channel, and then see if your Tor traffic slows down as a result.
This attack might be possible even if you aren't running anything you would think of as a network application -- he can send a lot of ping packets in your direction, and they'll congest your Tor traffic because your Tor flows share the same Internet connection as non-Tor traffic.
I should point out that the attack is still mostly theoretical: you'd have to get some really accurate timing results before you could draw strong conclusions from it. So in summary, the short answer to your question is "no."
If the attacker has some hint that you might be the one running some service (maybe because you mentioned it over unencrypted irc/mail/..), they can quite easily test/confirm it by flooding/killing your host. If the Tor stream they try to analyze and your service take the same hit at the same time, it's confirmed.