No. Absolutely and periodically not.

The North Atlantic Treaty Organisation was founded in 1949 (April 4th), just as the Cold War was kicking off and just as tensions between Soviet Russia and the US were rising.

For obvious reasons, Russia wasn't invited to be in NATO nor did it particularly want to, being an enemy of the west and all that. Instead, in 1955 the USSR formed the Warsaw Pact, which was effectively the Russian version of NATO. It was made up of Russia and seven other satellite states; Bulgaria, Czechoslovakia, East Germany (withdrew in 1990), Hungary, Poland (withdrew in 1990), Romania and Albania (withdrew in 1968).

With Russia's answer to NATO formed, they had no desire to join NATO and none of the member nations could really do so either. NATO is a superpower of its own though, and now contains a majority of the Warsaw Pact member states after the dissolution of the organisation in 1991, following the downfall of many of the Communist governments in the member states.

I hope this covers it for you, I'd be happy to answer any other questions about NATO and the Warsaw Pact.