I broke a bone in the top of my hand long time ago, but my hand is still swollen. Is this normal?

Yes. Most of the time it is do to new bone forming at the fracture site and you may always notice it. If it is still painful i would have your doc look at it.
How long ago. How long is "long time ago? Weeks, months, years? Get checked again. Not normal; can be poorly healed fracture, tendon, tendon sheath problems.