Not always. God comes first. He is above all. Let's examine a few key sections of Romans 13:1-4:
"Let every soul to the higher authorities be subject, for there is no authority except from God, and the authorities existing are appointed by God,
2 so that he who is setting himself against the authority, against God's ordinance hath resisted; and those resisting, to themselves shall receive judgment.
3 For those ruling are not a terror to the good works, but to the evil; and dost thou wish not to be afraid of the authority? that which is good be doing, and thou shalt have praise from it,
4 for of God it is a ministrant to thee for good; and if that which is evil thou mayest do, be fearing, for not in vain doth it bear the sword; for of God it is a ministrant, an avenger for wrath to him who is doing that which is evil." (YLT)
The civil authority is to be a ministrant for God, to do good works, to judge righteous judgment, to uphold God's word.
"10 And now, O kings, act wisely, Be instructed, O judges of earth,
11 Serve ye Jehovah with fear, And rejoice with trembling." (YLT)
"God reigns over the nations; God sits on His holy throne.
9 The princes of the people have gathered together, The people of the God of Abraham. For the shields of the earth belong to God; He is greatly exalted." (NKJV)
Psa. 148:7, 11;
"7 Praise ye Jehovah from the earth,
"11 Kings of earth, and all peoples, Chiefs, and all judges of earth,"
The civil rulers answer to God, and they are to be ministers of good for God. The definition of "good" is of and pertaining to God, what originates from God (Strong's Greek 18). They cannot do evil and at the same time be ministers of good things of God.
The Bible has to be taken as a whole, and one or two verses may not be used out of context to justify a particular idea or desire of man.
When the civil rulers and authorities pit themselves against God, we are not bound by their ungodly laws.
"Then Peter and the other apostles answered and said, We ought to obey God rather than men." (KJV)
The common argument that we must obey every law of any civil authority is trotted out by tyrants to excuse their evil deeds. They are not allowed by God to run rampant, to exercise unrighteous judgment, nor to oppress the people. Solomon's prayer in Psa. 72 was to be able to rule and judge with the righteousness from God.
Whenever any civil rule or law or code defies God we are not obligated to follow it.
Look at the lesson Pastor Chuck Baldwin brings on Romans 13 here.