It’s quite a common process known as a superinfection or secondary infection. Typically a bacterial infection occurs on top of a viral one. The mechanism in something like a cold is that the virus causes congestion, increased mucus secretions, and just generally a bunch of gunk which doesn’t get cleared very well. That acts as a nice breeding ground for bacteria which can now gain a foothold.

Typically that bacteria is just from your own skin and is normally harmless when kept in check by the other skin bacteria around it, but it gets a new advantage from the previous viral illness and grows unchecked.