I lowered the action on my old guitar and now some frets don't work. When I try play them it sounds as though I'm not fully holding in the fret down, but I am. What could be causing this?

it is a fairly basic guitar.
your frets don't work?

i think what you're trying to say is, you're getting fretbuzz. thats cured by lifting the action a little bit, all you've done is lower it a bit too much.
I thought it might be that but I had thought the action on my old guitar was higher than my new one. However the action on my old guitar does seems a little lower.

It's not really buzzing its just you know the sound it makes when you aren't fully holding in the fret? It sounds like that.

I'll just put the action up a bit.

Lowering the action with out checking the neck might be the problem you probably need to do a truss rod adjustment. The other problem could be the frets are not level some might be higher or lower than others causing dead spot on the neck. Usually when the action is high you do not notice these problems but lowering the action can bring them to you attention fast. You should take it to a guitar tech and have it checked out.

All guitar are different depending on the neck, action and frets what works on one might not necessarily work on another.