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1.
found usually on blackberry curve (non-3G) devices when trying to login to twitter or other apps, especially in slow internet connection/EDGE areas.

It is a misnomer why this error message is displayed and not simply: no connection to server. Perhaps reference is being made to the particular API (-set of programming instructions) that allows twitter and other apps to connect.

It is thought by many to be a product of an under-developed device and/or network.
Blackberry screen error: "Twitter failed to connect. You have reached your API limit."
by Farcry7 July 04, 2011