St. Cyprian of Carthage was very supportive of Rome when it worked to his favor, but when he disagreed with Rome he had no problems saying outright that Rome was wrong. So, for example, when Pope Stephen of Rome and St. Cyprian disagreed about the validity of certain baptisms, St. Cyprian arranged a council and opposed the views of Rome. Rome fired back by condemning the decisions of the Council. St. Cyprian then responded by holding another Council, with more bishops, reaffirming what they had decided at the previous council.
Of course many examples that can be brought forth are of this type, because while there were claims from early on (especially from mid-5th century), it would be anachronistic to read modern papal claims back into the ancient texts and actions. Rome sometimes got involved in the affairs of others just like other sees did. For example in the late 4th to early 5th centuries, Alexandria got involved with the affairs of Constantinople several times, often having a significant impact (for better or worse). It's hard to pick apart what was Rome's genuine belief about it's role, and what the genuine belief of others about Rome's role.