The Dropbox API does have a general rate limiting system that applies to all account types, but we don't have any specific numbers documented. It operates on the basis of calls per period of time. It generally works on the scale of seconds to minutes though, not days to months.
Also note that not all 429s and 503s indicate explicit 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.
Additionally, there is a specific rate limit that applies to upload calls for some Dropbox Business plans only, which does operate as a per-month limit. You can find more information on that here.
I recommend referring to the error documentation and Error Handling Guide for more information.
