"Omnipotence" doesn't mean that God can do what is logically impossible. The oft-repeated question, "can God make a square-circle?" can be confidently answered by the classical theist with a "no". God's being omnipotent means that He can do all things, but square-circles aren't things, so God's inability to create them has no effect whatsoever on His omnipotence.
This is where Judges 1:19 comes in. It is commonly alleged that this passage is inconsistent with the doctrine of God's omnipotence. Here is the verse:
"The LORD was with the men of Judah. They took possession of the hill country, but they were unable to drive the people from the plains, because they had iron chariots."
Does this verse imply that God is (or was) incapable of overcoming iron chariots? Obviously, such a feat is logically possible for God, so the claim is that God isn't omnipotent according to Judges. However, does this brief interpretation of the text do justice to the author's original intention?
I should point out that, just intuitively, this seems to be an extraordinarily implausible interpretation. Are we really to believe that God, whom Judges says that mountains quake before (Judges 5:5), the One who brought up Israel from the land of Egypt (Judges 6:8) is really inferior to the chariots of men? Presumably this is what the proponent of the argument is suggesting.
Now, we have at least two ways of responding to this, while maintaining that God is omnipotent. First, we could simply say that the Bible is mistaken on this point. "The arguments of natural theology suffice to show that God exists and is all-powerful, even though the Bible is fallible." This kind of response can be modified like this: "Even if this is what the Bible teaches, God still exists and is all-powerful. At the most, this only implies that the Bible is mistaken on this point."
The above reply, in its latter form at any rate, is correct. Moreover, it is a decent polemical response. Nevertheless, being the Biblical Inerrantist that I am, I believe the Bible is correct whilst maintaining that God really is omnipotent. So, how do we reconcile these? As they say in real estate: the three most important terms are location, location, location. Well, for any interpretation of the Bible, the three most important terms are context, context, context.
For starters, let's take a look at the subsequent text, Judges 1:22-36:
Now the house of Joseph attacked Bethel, and the LORD was with them. 23 When they sent men to spy out Bethel (formerly called Luz), 24 the spies saw a man coming out of the city and they said to him, "Show us how to get into the city and we will see that you are treated well." 25 So he showed them, and they put the city to the sword but spared the man and his whole family. 26 He then went to the land of the Hittites, where he built a city and called it Luz, which is its name to this day.
27 But Manasseh did not drive out the people of Beth Shan or Taanach or Dor or Ibleam or Megiddo and their surrounding settlements, for the Canaanites were determined to live in that land. 28 When Israel became strong, they pressed the Canaanites into forced labor but never drove them out completely. 29 Nor did Ephraim drive out the Canaanites living in Gezer, but the Canaanites continued to live there among them. 30 Neither did Zebulun drive out the Canaanites living in Kitron or Nahalol, who remained among them; but they did subject them to forced labor. 31 Nor did Asher drive out those living in Acco or Sidon or Ahlab or Aczib or Helbah or Aphek or Rehob, 32 and because of this the people of Asher lived among the Canaanite inhabitants of the land. 33 Neither did Naphtali drive out those living in Beth Shemesh or Beth Anath; but the Naphtalites too lived among the Canaanite inhabitants of the land, and those living in Beth Shemesh and Beth Anath became forced laborers for them. 34 The Amorites confined the Danites to the hill country, not allowing them to come down into the plain. 35 And the Amorites were determined also to hold out in Mount Heres, Aijalon and Shaalbim, but when the power of the house of Joseph increased, they too were pressed into forced labor. 36 The boundary of the Amorites was from Scorpion Pass to Sela and beyond.
In verse 22, we again find the phrase, "and the LORD was with them," which connects this passage with verse 19 above. Following this, we see that although the Israelite tribes conquered the land to an extent, they did not obey God entirely. The Canaanites and others who were living in the land remained in the land, which God forbade the Israelites from allowing (which is apparent from the text above, as well as Deuteronomy 7:24 and Joshua 9:24, among others).
The implication here is that although God was with them, Israel didn't do what He commanded them to do. As a result, Numbers 33:55-56 was fulfilled:
"'But if you do not drive out the inhabitants of the land, those you allow to remain will become barbs in your eyes and thorns in your sides. They will give you trouble in the land where you will live. 56 And then I will do to you what I plan to do to them.'"
Throughout the book of Judges, the people of Israel are raided, invaded, and conquered by their enemies. Whenever they return to God, God saves them. It was Israel's lack of faith and disobedience that is responsible for their inability to overwhelm the iron chariots of Judges 1:19, rather than some defect in God's power. This, I think, is a plain reading of the text, which allows the Bible to speak for itself.
We can see Judges 1:19 as a parallel to Numbers 14, in which the men of Israel scout the land of Canaan. They see that the men of Canaan are strong and well-armed, and they doubt they can defeat Canaan. God is displeased by this and commands them to go into the wilderness as discipline. The men of Israel regret their lack of faith and, disobeying God yet again, decide to invade Canaan. The invasion is predictably unsuccessful. Why? Because they did not obey God's command. As Matthew Henry so ably summarized three hundred years ago: "The Canaanites had iron chariots; but Israel had God on their side, whose chariots are thousands of angels, Psalm 68:17. Yet they suffered their fears to prevail against their faith." (Emphasis added).
Pointing to Judges 1:19 may provide the skeptic with an additional reason for unbelief, but fortunately for us, their interpretation of the text is left wanting. After all, it is said that they (referring to Israel, not God) were unable to drive the people from the plains.