There is no real guideline to provide. The "correct" number is a function of hardware, traffic, and the amount of processing needed to fulfill requests. That number may change over time as the app is further developed.
Increase the number of available processes/threads across some amount of time (perhaps a few days or a week to account for bursts). Be sure to monitor CPU, RAM, and disk I/O. If any of those reach 100% utilization during regular operation (or your response times drop), back it off slightly to account for bursts. Keep in mind that you may also need to tune the number of database listeners or connections, and/or RAM allocated to the DB cache, etc. Other dependent pieces of your infrastructure may need modification as well, such as caches.
No bad side effects until you exceed RAM, CPU, or disk limits. In many cases, your DB is the bottleneck, so be sure to tune those settings along with the web service processes.
It's a bit strange to receive a 403, since that generally has to do with authorization, and not operational issues. Seems like a 5XX error would be more appropriate for a ChannelFull situation, but that may be just the way it works.
-James