Can an app mess up your phone?

Yes.

yes, some bad coded apps can slip through the cracks

Steveo47 said:

Is it possible For an app off the app store (not cydia) to slow down your iPhone 4?

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If you explain a little more I can give you a precise answer.

Apps usually can't harm when they are not running. Even when in background (since they are actually not really running) except if it's getting GPS events or playing audio.

There are other things an app can do that will slow down the phone but they should be easy to see; for example an app can add 100,000 contacts to your address book; when you use the address book it will be slower but you should see the contacts...

Lastly there are things that may happen that YOU may think are caused by an app (maybe you have just finished using it) but really depends on other factors.

Steveo47 said:

Well I'm just askin because I had an instance where my safari froze, as a result I had to quit safari and redraft. And another instance where my entire phone froze quitting all the apps in my multitasking bar. It froze when I tried to close the "notes" program. I'm not sure if that's normal or what

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Safari had a couple issues on my phone also, it could be a bug. I don't think what happened to you can be the cause of an app, seems more an issue of the iOS.