UPS's are notorious for nuisance tripping GFCI's due to the extensive power filtering they usually have inside. It may not happen right away but is dependent on how 'dirty' your power is and on how sensitive your particular brand of GFCI is. All it may take is a neighbour using his AC or power saw and it could generate a spike large enough to knock your system offline (UPS filters spike to ground, leakage current trips GFCI which needs to be manually reset, UPS runs out of battery because no one's home to restore power).

This advice is specific to the US & Canada where GFCI's are calibrated to trip at 5 mA. In the UK, their RCD's trip at 30 mA so it's less likely.

If you read the fine print for your UPS and/or GFCI they will say not to use them together.

What about AFCI's? Well they are basically nuisance devices by design and most electricians hate them. Do not use unless absolutely necessary by code.

The difficulty comes into play because UPS's, servers, etc are essentially industrial equipment while the code is written for such common residential usages as plugging in a lamp (where a dog will chew on the cord, a good way to start a fire hence a perceived need for AFCI protection.)

What if you're stuck because you want to locate your UPS and equipment in an area where GFCI's are required (like a residential basement)? Well I won't tell you what to do in your own home but I'm sure you can devise a creative solution. :)