1996 Chevy Cavalier Ignition Won't turn.

Try a little lock-eze in the cylinder. The tumblers are binding...

Tumblers gets sticky from harden liquid from key that have too many dinners. You might want to avoid spilling Pepsi on them.

Pushing keys in deeper just complicates the matter. This is the reason why we avoid liquid/paste lubricant for them. Use graphite lubricating powders.

Second scenario: Keys probably roughly cut. You can try paper sanding the keys down to give it nice, gentle curves. The smoothness allows the tumblers to move across the keys with ease. And some key cylinders can be removed for cleaning.

-- Milo