When I flush the upstairs toilet, after about 30 seconds, my downstairs ceiling begins to drip of water. What could cause this situation? Is it because the toilet is partially clogged? The toilet appears to be working fine.

Could be a bad wax ring or a leak in the drain pipe. You must open the ceiling any way and once you do you will see the leak. When you have done that come back and we will help fix the leak and even repair the ceiling. Is it drywall or what?

If the toilet flushes properly, you probably have a bad wax seal or a cracked or broken flange. In some older houses, the lead toilet flange will disconnect from the cast iron hub. Is the toilet tight or does it rock back and forth? Could be a cracked pipe also but I'd go for a new seal. Need to remove the toilet and look in the pipe. See any cracks? If not replace the seal. $10 DIY fix.