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I have the exact same problem on iPhone 4 / iOS 6.1.3
(and just discovered the "mee too" button - so please don't flame me for posting this. Cannot delete posting.)
Posting is good - no need to apologise. Try this and shout if you still have problems.
Hold the Home and Lock buttons together for 10 seconds to restart your device.
-- Connect to a different network and see if the problems persist.
-- Perform a clean reinstallation.
Actually I just powered down my iPhone - and then turned it back on. After that the Spotify App can start successfully and I can access my account.
Great - enjoy your music 🙂
Rebooting the device did the trick. Though, i reinstalled the app yesterday and lost my 700 songs list with "offline mode" and now i have to resync all the songs to make them offline available 😞