The Dropbox API does have a rate limiting system, but we don't have specific numbers documented. It is only designed to prevent abuse though, and is accordingly relatively generous. Further, the limits operate on a per-user basis. That being the case, you generally don't need to worry about hitting it in normal use.
Also note that not all 429s and 503s indicate rate limiting, but in any case that you get a 429 or 503 the best practice is to retry the request, respecting the Retry-After header if given in the response, or using an exponential back-off, if not.
(Note, for Business accounts, there is a data transport limit that depends on the team's plan:
https://www.dropbox.com/help/9291 )
