The election of Abraham Lincoln in 1860 was really the last straw for the Southern states. His election, along with the rise to power of the Republican Party, spelled doom in the minds of Southerners. Although Lincoln would probably not have pressed for an end to slavery immediately had there been no secession or civil war (his later Emancipation Proclamation was really as a result of the war), Southerners believed that Republicans would eventually abolish slavery or, at the very least, allow no more slave states to be admitted into the Union. This would have greatly decreased the Southern states' power in the Senate and House of Representatives, and their representatives would have found themselves in a distinct minority in other issues aside from the slavery debate. The slavery issue was the South's foremost concern, however, and they could not envision a South without slave labor for the harvest of "King Cotton" as well as other food crops and tobacco.
There are a number of reasons why this was the case.
First, all sorts of compromises had already been tried and had failed to end sectional tensions. This meant that there were not really any more ideas out there for how to create another compromise.
Second, this first problem was partly due to changes in law. The Dred Scott decision had made it so Congress could no longer make laws about slavery in the territiories.
Third, the North and South had completely run out of good will towards one another. As the years went by and various fights over policy had occurred, the North and the South came to trust each other less and less. Compromise is more possible when some level of trust exists between the two sides. By 1861, there was no trust left, only animosity.
For these reasons, compromise was no longer possible by 1861.