All true authority comes from God (John 19:10, 11). That is a fundamental principle. Another is: God must be obeyed rather than man when the two conflict (Acts 5:28, 29). Rulers had no right to forbid what God had commanded (Acts 5:20). If a clear command of God is forbidden, it must not be obeyed regardless of consequences. But many commands are not clearly opposed to one another. What then?
First, recognize that there is a difference between Romans 12 and 13 (the former having to do with how individuals relate to one another and how Christians must relate to the government (see also 1 Peter 2:13, 17). Failing to recognize this distinction can cause confusion. New Testament Christians had to live in countries hostile to Christianity as many do today. Some went to the stake for living as God requires (martyrs). Others (confessors) stayed true to God but survived. Some of the latter did so by observing other biblical injunctions. They were conscientious in paying taxes, for instance (as Jesus was: Mark 12:14ff). They provoked no trouble if it was possible to avoid it (Proverbs 14:16; 27: 12). This warning is important to apply to many situations today.
Another principle can be deduced from Jesus’ command to his followers who were to serve Him. If you don’t have a sword, sell your coat and buy one. A coat was important—people slept in them (if taken as collateral, they had to be returned by night). Jesus’ command was to be prepared to defend yourself if attacked (Lk. 22:36). Rome had cleared some of the roads of thieves, but not all of them. Because of government failure, people had to take the law into their own hands when it was necessary to do so. This could be the most important principle of all in many situations—but it’s application must be thought through carefully (and no fudging is allowed!). The government is not the enemy unless it makes itself so—Paul often availed himself of its authority—and had friends in it (Acts 19:31). It would be interesting to hear what other believers faced with authority issues have done.