It does, to some extent, depend upon the manufacturer of the UPS system and how it is setup on the site.
Some manufacturers have an algorithm within the software that adjusts the load capacity of the UPS depending upon ambient temperature. At 40 degrees C the UPS will be rated at its nominal load, whilst at 25 degrees C the inverter is able to supply 110% of In indefinately.
For higher overloads the system can remain on-line depending upon the duration of the overload, for example:
125% load - 10 minutes
150 % load - 10 seconds
200% load - 1 second.
The UPS voltage and frequency will be stable during these overload events (at least for a decent UPS) and the limits are set for thermal considerations of the output switching devices and magnetics. At the end of these periods the UPS will then switch to the bypass (reserve) supply if this is available. If not then the inverter will shut down and load will be lost.
As mconroy mentions, most UPSs also have the facility to current limit the incoming rectifier current. If the nominal current of the rectifier is exceeded then the rectifier will lower its voltage until current is being drawn from the battery bank. As the batteries discharge the rectifier voltage will keep going down until the minimum battery voltage is reached and again the inverter will shut down.
Some UPSs also have the facility to current limit the output of the inverter for situations where the Bypass isn't available and an extremely high load (>300%)or a downstream fault is applied to the output of the UPS. In this case the output voltage will fall dramatically until the fault is cleared or the overload removed.