I've just mounted a 42" TV with a new wall mount that uses 3/16" bolts into some studs on the wall. I now want to use an articulating one, which means I'll need to replace the old mount. The new mount also uses the same size bolts. Is it safe to re-use the holes from the old mount with the new mount, or should I drill new ones?
If you didn't strip the holes when removing the old bolts, then you PROBABLY should be able to re-use the same holes, but to be safe, I would inject a little Elmer's wood glue (just a few drops) into each hole before you insert the bolts. This is just to reinforce the wood in the hole and help ensure it stays locked in place.
But honestly, I think you'd be wise to drill new pilot holes 1" above or below the old ones, and fill the old ones in with wood putty capped w/ a little dab of plaster (or toothpaste). The old holes will end up hidden by the mounting plate anyway, and better safe than sorry.
Put a dab of wood glue in the hole and push it all the way in with an appropriately sized nail, and then jam wood-glue-coated toothpicks into the hole. You might need to give them a tap with a hammer to get them all the way in.
Give it 30 minutes to harden, and then you're good to go.
One trick I've seen someone do when reusing a hole is to put a small piece of copper wire in there so the screw has something to tighten up against.
If the screws go back in and hold tight, it should be OK, however, the worst case scenario of a TV falling off the wall is pretty bad, so sometimes it is better to be safe.
The best solution is to drill new holes a couple inches away from the existing holes. If space and positioning allows, you can even re-use one row of the existing holes and drill new holes for the other row of screws (most brackets have 2 rows of holes).
If you are set on re-using the same holes, I would suggest that you either use a lag that is 1/2" longer than the previous lag (so it grabs new wood at the back of the stud; make sure its not too long or you might be in for a surprise on the other side of the wall), or use a lag that is one size larger in diameter.
I don't love the glue/dowel/toothpick solution. It might work on door hinges, but you are now putting all of your money on the strength of the glue and toothpicks - neither of these were designed to hold large loads.
I always hang on a TV mount with as much weight as I can to ensure it is secure. If it will hold me, it will hold a TV! If it comes off the wall then better that it happens without an expensive TV attached and a child under it.