Can a Pokemon get a status condition of the same type?

I know the title might sound a little confusing, so let me give a couple of examples.

If a pokemon is a poison type, can it be poisoned? If a pokemon is a fire type, can it be burned? This also goes for the other types; electric > paralyzed, ice > frozen. And of course if I'm missing any others.

In my experience of playing the games, I don't recall a time this ever happened to me.