It's more likely the power supply in the system than the UPS, actually. PSU (power supply unit, as opposed to UPS, which is uninterruptible power supply) problems are some of the most difficult to diagnose for someone that hasn't dealt with them many times before. Variation in power output from a PSU can cause all kinds of weird, unpredictable problems, and can cause gradual degradation of performance and functionality of other components.
Power supply problems are one of the most serious, and one of the most elusive, hardware failures, because the symptom set is so hard to define and the problem can cause more problems in the rest of the hardware.