Not sure about developing a cold but if you look at CMV (cytomegalovirus) and EBV (Epstein Barr virus), those are both prevalent in most people as latent infections. If you get another infection, are immune suppressed, or pregnant, then those viruses can flare up and cause their relevant symptoms.

Probably the most well known is glandular fever which is caused by EBV. You can have it, then recover, bit still have dormant/latent virus in your system that flares up again in the future. Same goes for TB