The word is at the beginning of this verse is not in the original Greek. This is significant, because the rest of the verse is entirely in the past tense: "had been," "given," "could have given," "should have been." Given that the context has been the Old Covenant, and not God's eternal codification of right and wrong, if a word needs to be added to make the meaning more clear, the word "was" fits the context better: "Was the covenant against the promises of God?"
God's promises to Abraham were not contrary to the covenant. God's promises to Abraham and God's covenant with ancient Israel accomplished different things! The Old Covenant was a temporary agreement, given until the Seed arrived. It had health, wealth, national strength, and military protection and dominance as its benefits. The promises to Abraham had physical components as well, but with the promise that includes eternal life (see the notes at Galatians 3:14) comes a plethora of other benefits. The promises and the Old Covenant were not in disagreement with each other, but rather the Old Covenant was an auxiliary to the promises, to further the plan inherent within the promises so God would have a nation to send His Son to in order to begin the church—the Body of Christ, which will become His Bride.
God's Law is an extension of His very character and nature. When He codified it and set it before Israel, one of the results was that it showed anybody who was willing to look that mankind does not naturally gravitate to God's way; it is exceedingly difficult to live God's way because our human nature wants to fight it at every turn. The Old Covenant demonstrated to Israel, and to anybody who reads Israel's history, that man needs a different heart in order to come into alignment with God's intent for him. It also demonstrated that the blood of innocent animals was not sufficient to really take away the sins of the people. Those animal sacrifices were only a shadow of the reality to come—Jesus Christ. The whole system was on a physical level, without spiritual promises. The Holy Spirit was not generally available, and so all of Israel's experience also demonstrates that without the right heart, without the same intent and desire as God, it is impossible to come into alignment with Him. And that new heart is only available to those with whom He makes the New Covenant.
God is faithful in His promises. For Abraham to have a "great nation" come out of him, and for him to have a "great name" through his posterity, and for him to be a blessing, God had to enact a system by which an unlearned people could live in accordance with the natural laws that would bring about the desired results. The Israelites, after living in Egypt—sin—for so many generations, had that sin implanted into their national character, and it was perpetuated by their generations even after they left Egypt. God made the Old Covenant because of their sins, to show them just how far off course they had gotten.
This covenant also was to be a preparation for Christ's first coming and the New Covenant. If they had used the covenant properly, it could have given them a head start when the Seed arrived, and they could have seen what God was working out. Paul shows in his writings to the Romans that the Jews (Israelites) had "first dibs" (Romans 1:16; 2:9-10), but that the opportunities were no longer exclusive to the chosen people. Paul also shows that the Old Covenant was a means by which to preserve the knowledge of God's laws in Israel to prepare them for the coming of Christ! In this way Christ could return to a people already familiar with His laws and statutes, rather than coming first to a Gentile nation that He would have to teach essentially from scratch.
When He came as a man, He came to a people and a culture that had a long history of at least familiarity with His laws, if not complete obedience. They clearly did not understand the true intent behind the laws, and even abused them to the point of thinking they could be justified by keeping them, but this way of life was not entirely brand new to them in the way it would have been if Christ first appeared in Africa or the Orient. There would have been no hope of those Gentile nations recognizing how Christ fulfilled and personified the law. Without the history of the Old Testament, Christ would have meant very little to them, although He would have undoubtedly been worshipped as a type of deity because of the miracles He performed. But He was not seeking worship—He was seeking a people who would live as He lives.