Yeah, cause its never happened to anyone else before...
Last fall, Apple released their App Store Approval Guidelines. The relevant guideline—the only place where the word "duplicate" appears in the guidelines—is quoted on Stackoverflow:
Apps that duplicate apps already in the App Store may be rejected, particularly if there are many of them, such as fart, burp, flashlight, and Kama Sutra apps.
If you were to write and submit your own app that connected to Dropbox, it might get rejected. Given the number of third-party Facebook apps and Twitter clients still available on the App Store, however, I think that unlikely.
Plus there's no no shortage of web browsers on the App Store.
I feel pretty good about Dropbox never being pulled for "duplicating functionality."