Q: Push notifications often do not alert as they should.

Hello DaemonPix,

Thank you for the question.  It sounds like there is you are not receiving notifications while your phone is locked.  Have you checked to see if Do Not Disturb was accidentally enabled?  You will see a "moon" icon in the status bar if it is enabled.

You can disable Do Not Disturb in Settings > Do Not Disturb or in the Control Center as described in the following articles:

iOS: Using Do Not Disturb

http://support.apple.com/kb/HT5463

iOS: Understanding Control Center

http://support.apple.com/kb/HT5858

Thank you for using Apple Support Communities.

Best,

Sheila M.