"Most assuredly we cannot understand the New Testament apart from the Old. The proof is simple: try reading Hebrews, Jude, or Revelation without any reference to Old Testament allusions or concepts. It cannot be done (which may explain why many modern Christians avoid those books, too).
The Gospels can be read as stand-alone stories, but a reader unacquainted with the Old Testament will miss many layers of richness in them. Paul constantly appealed to the Old Testament. Without exception, every New Testament author wrote about the new work of God on earth while looking through the prism of the earlier or "old" work.
A Chinese philosopher insisted on riding his mule backward so that he would not be distracted by where he was going and could instead reflect on where he had been. The Bible works in somewhat the same way. The Epistles shed light backward on the events of the Gospels, so that we understand them in a new way. Epistles and Gospels both shed light backward on the Old Testament. For centuries, the phrase "As predicted by the prophets" was one of the most powerful influences on people coming to faith.
Justin the Martyr credited his conversion to the impression made on him by the Old Testament's predictive accuracy. The brilliant French mathematician Blaise Pascal also cites fulfilled prophecies as one of the most important factors in his faith.
Nowadays, few Christians read the prophets except in search of Ouija-board-like clues into the future. We have lost the Reformers' profound sense of unity between the two testaments.
Understanding our civilization and understanding the Bible may be important reasons for reading the Old Testament, but perhaps the most important reason is this:
It is the Bible Jesus read.
He traced in its passages every important fact about himself and his mission.
He quoted from it to settle controversies with opponents such as the Pharisees, Sadducees, and Satan himself.
The images–Lamb of God, shepherd, sign of Jonah, stone which the builders rejected–that Jesus used to define himself came straight from the pages of the Old Testament.
Once, a government tried to amputate the Old Testament from Christian Scriptures. The Nazis in Germany forbade study of this "Jewish book," and Old Testament scholarship disappeared from German seminaries and journals. In 1940, at the height of Nazi power, Dietrich Bonhoeffer defiantly published a book on the Psalms and got slapped with a fine.
In letters of appeal, he argued convincingly that he was explicating the prayer book of Jesus Christ himself. Jesus quoted often from the Old Testament, Bonhoeffer noted, and never from any other book. Besides, much of the Old Testament explicitly or implicitly points to Jesus.
The Old Testament contains the prayers Jesus prayed, the poems he memorized, the songs he sang, the bedtime stories he heard as a child, the prophecies he pondered. He revered every "jot and tittle" of the Hebrew Scriptures. The more we comprehend the Old Testament, the more we comprehend Jesus.
Said Martin Luther, "the Old Testament is a testamental letter of Christ, which he caused to be opened after his death and read and proclaimed everywhere through the Gospel."