Can a game that's purchased by one psn account be playable by another pan account?

My girlfriend bought Minecraft on her her psn but when I go to start up the game on my own it says I don't have the license, is there a way to fix this?

Yes as long as the psn account that it was downloaded on is entered into the concel that you have and then downloaded it will work

What your trying to do is called gameshare. Follow these simple steps
1. Make a new PSN user
2. Sign into Your girlfriend's account but don't save the password or have auto sign in on
3. Go to account management
4. Go to activate system and activate your PS3
5. Go to transactions and go to the downloads and download the game you want or if you downloaded it already then it should work now.
5. Minecraft should work now
6. I'm glad to help