Most modern scholarship suggests that Southerners did not chafe at the use of federal power, but that it would no longer be used to further their agenda. They had no problem in using federal power to force slavery on the territories in violation of popular sovereignty (as they did with the so-called Lecompton Constitution in Kansas), in subjecting Congress to "gag rules" concerning the debate over slavery, and indeed enforcing fugitive slave legislation in violation of local personal liberty laws in the North. The idea that "states rights" were actually an issue is more related to Lost Cause ideology than modern historical work. For a good synthesis of the recent historiography of the political crises of mid-century, see William Freehling's Road to Disunion or Allan Tulloch's History of the Civil War Era.
The sectional differences between the north and south were kept at bay by the compromises of 1820 and 1850, and were transitory at best. While both compromises were aimed at maintaining the balance of power in the Congress, the underlying issue of slavery intensified throughout the 1850's polarizing the north and south. The causes responsible for the disintegration of compromise were:
1. the continued rise of the Abolitionist movement
2. the anger from both sides regarding the Fugitive Slave Act; the north because it was passed, the south because many in the north ignored it
3. the publication of Uncle Tom's Cabin
4. the deadly result of the Kansas-Nebraska Act 1854
5. the new 'Republican' party whose platform was to prohibit slavery into the west
6. the decision by the Supreme Court in Dred Scott v. Sanford 1857 which ruled that slaves were property, nullifying the Missouri Compromise and any other ruling regarding slavery
7. John Brown's raid on the Harper's Ferry arsenal in 1859
The balance of political power in Washington had been dominated by the "Free State, Slave State" balancing act that started right from the founding of the country (Look up the acceptance of Vermont and Kentucky as state numbers 14 and 15, one slave, one free, one North, one South.) As the country expanded, it was necessary to compromise on the issue, as in 1820 and 1850, but by 1860 it became increasingly difficult to maintain the balancing act.
As others have posted, it wasn't just about free and slave states, it was about the differing cultures that developed in the North and South and the different ways by which they wished to be governed.
Of the posts above, Post 3 is the best. It is right to look at the importance of social and economic differences between the two sections of the country. We should also note that the reason that things got worse by 1860 was because there had been fairly constant friction over the past 40 years with regard to slavery. This led to a complete breakdown in trust between the two sections by 1860.
Basically, compromise failed because Southerners perceived, with the rise of free soil ideology and the Republican Party, that they would no longer be able to use the federal government to promote the spread of slavery in the West. It was ultimately an unwillingness to compromise on the part of the South, which became manifest after 1850, that caused the split, not anything inherent about a two party system. The Republican Party itself was the product of a series of compromises, between abolitionists and Free-Soilers in particular.
The cultural and sociological divide between the industrial North and the agraian South led to the divisiveness of ideologies in the 1860s. Certainly, there was some economic envy of the free labor and wealth that was possessed by plantation owners who lived virtual aristocratic lives. On the part of Southerners, there was much resentment of the controls that the Federal government, located in the North, wished to place upon them.
The two-party system does not seem to invite compromise. At that point in our political history we had more parties involved, and more compromise possible or at least attempted. Two ideologically opposed parties cannot compromise, apparently. That is our problem now.