Power supplies can test fine but still be bad

Today I witnessed an attempt to revive a dead machine, the one that failed suddenly several months ago. The original motherboard had died, so a new one was installed. Since there had also been concerns about the power supply, the new motherboard was first tested with a different computer's supply, resulting in a successful POST. When the machine's original power supply was tried, the CPU briefly spun up, but then went back to doing nothing. This was bizarre because both power supplies had previously tested as just fine. Evidently a power supply can provide the right voltages to a tester but still fail under real loads.