As in, if I want to catch and notify an operator of all high severity errors 17 through 25, all I needed to do was set up an Alert for Severity 17. This is false.
Here's a quick proof:
As you run the above script statement-by-statement (NOT in a batch), you'll see the occurrence data trigger only for an error with the severity EQUALS the defined alert state. In this case, the Severity 21 error did NOT trigger the Severity 20 alert.
Make sure that database mail is setup, that SQL Agent is configured to use the appropriate Database Mail profile, and that the failsafe operator is enabled to email your DBA team.
Error severity info: http://msdn.microsoft.com/en-us/library/ms164086.aspx