Can you put a +1/+1 counter and a -1/-1 counter on the same creature and do they cancel out?

Both +1/+1 counters and -1/-1 counters can exist on the same permanent at the same time — nothing prevents them from being placed there. However, the game's state-based actions immediately remove them in equal pairs. This is covered in CR 704.5q, which states that if a permanent has both a +1/+1 counter and a -1/-1 counter on it, that many of each type are removed until only one kind remains (or none remain).

State-based actions are checked continuously and happen before any player gets priority (CR 704.3). So the cancellation is nearly instantaneous — no player can respond to the counters existing together, because state-based actions fire before anyone gets a chance to act.

The cancellation is always one-for-one. If a creature has three +1/+1 counters and two -1/-1 counters placed on it, the state-based action removes two of each, leaving the creature with one +1/+1 counter and zero -1/-1 counters.

Concrete example: You control a 2/2 creature with one +1/+1 counter on it (making it effectively 3/3). An opponent casts a spell that puts two -1/-1 counters on it. State-based actions then remove one +1/+1 counter and one -1/-1 counter, leaving the creature with one -1/-1 counter — so it is now a 1/1.

This interaction is relevant with cards like Devoted Druid (which untaps by putting a -1/-1 counter on itself) and proliferate effects that add +1/+1 counters, since the counters will simply cancel rather than stack independently.