Not knowing your psu tester, I can't say.
It is not clear to me if you have connected the leads that your tester can test correctly.
If a psu tester fails a test, then I would assume the psu is bad.
But a psu tester only tests basic voltage outputs and says nothing about the ability of the psu under load.
Your psu may fail under load if it is a poor quality unit that should not be used regardless.
Exactly what make/model psu do you have?