Wouldn't that depend on how comprehensive the background check was, and what items on the background checks showed up?
For example, I suggest the following should rule a person out from being allowed to own or use firearms
1. Any criminal conviction
2. Any conviction for drug use or possession
3. Under any legal restriction such as a trespass, stay-away or restraining order
4. Any conviction for a breach of firearms laws
5. Any conviction for domestic abuse
Under that list, 35% to 62% of spree shooters would not have been allowed to own a gun.
http://www.internationalskeptics.com/forums/showpost.php?p=12784068&postcount=62