I know what you mean! Unfortunately, Federal Law mandates that the FCC run these Alerts. What is the alert you are getting?
Amber Alert (Missing Child)?
Emergency Broadcast Test?
Those things like to embed themselves in the system and can stay for as long as 20 minutes! UGGHH. Generally they are gone by the time the response has rolled around a second time. A warm reboot often will not solve the problem. You often just have to ride out the alerts with time.
I have never tried a cold reboot to get rid of an Emergency Alert, That might work because it will clear the system of any data. This is done by unplugging the system for about a minute, and letting the thing reboot when you plug it back in.
You can always call and complain about the Alerts. If your box does not make the alerts go away within 10 minutes, ask them to submit a bug report to the Engineering Department about the specific issues your box has when dealing with Emergency Response Alerts.
I have actually seen a Navigator specs sheet that talked about a bug fix relating to an Emergency System Response Screen Alert, so TWC will take a bug report, but you must specifically request that the problem that you have be forwarded to the Engineering Department. Those alerts should be gone within 5-10 minutes of initial duration.