Is it possible to collide planets?

Solution 1:

Planets and moons are "on rails". You can not affect them or their orbits with the physics engine.

However, you can change their orbits with the mod Hyperedit. You can also use this to put two planets on a collision course, but the result won't be what you would expect. The game engine is not designed to handle planetary collisions. The planets will just move through each other without harm while any crafts on the planets or in their orbits will encounter some weird glitches.

Solution 2:

Scott Manley tried to move Gilly in one of his videos. He said "no" in the first 30 seconds and then spent the rest of the video showing what it would require if it were possible.