Of course! One of my favorite examples was the Nullification Crisis that happened in the early 1830s.
As the population and geography of the US was expanding in the 18-teens and 1820s, roads, canals and bridges needed to be build to facilitate transportation and trade over longer distances. Congress needed a bunch of money to complete these projects - it took a lot of men to break up the ground and remove obstacles like roots and rocks. Congress's favorite source of revenue was tariffs (import taxes) and to fund the projects like the Erie Canal and Cumberland Road, they raised tariffs again and again. The breaking point came with the Tariff of Abominations in 1828. Similar to other issues preceding the Civil War, this was a sectional issue. Northerners liked tariffs because a manufacturing economy existed there. Import taxes make foreign goods expensive and keep domestic products cheap. So if a Northern manufacturer wanted to have a competitive advantage over, say, British manufacturers, tariffs were great for his business. Southerners, on the other hand, disliked tariffs. Many of them had no trouble selling their cotton or food, and so tariffs didn't protect their businesses. The only effect tariffs had on them was that some stuff became steadily more expensive, and tariffs were viewed as a nuisance in the South.
The Tariff of Abominations (it was called that because it was so high, Southerners really hated it) was a tipping point for Southerners, who felt like continually raising tariffs was a sign that Northerners were in control of Congress. John C. Calhoun had recently (in 1826) published the States Rights' Doctrine, which asserted that states collectively had more power than the federal government and that if Congress enacts laws that are unfair to a particular state, such as South Carolina where Calhoun was from, that South Carolina should be able to nullify the law within its borders. Robert Y. Hayne then argued in Congress that South Carolina was unwilling to collect the import taxes in South Carolina because they were unjustly high and unfair to Southern planters. Daniel Webster, a northerner argued that South Carolina could not veto federal laws on behalf of any state, including itself - nothing would ever be passed in Congress or enforced within any state. He also argued that the Supremacy Clause of the constitution states that the Federal Government has more power than any specific state, so South Carolina must comply.
When Andrew Jackson got involved, he urged congress to pass the Force Act so that he could use the military to force South Carolina to collect the tax. When that law was passed, Southerners had a lot of motivation to find a way to compromise on tariffs. Henry Clay (Kentucky) devised a tariff that remained high in the short term, but would decline over time as the road projects would eventually dwindle and funding would not be needed for them. This compromise ended the nullification crisis, but set the stage for future debates about states' rights that wouldn't be settled until the Civil War.
There's obviously nothing on the scale of the Civil War, but conflicts between major slaveholding states and free or much less slaveholding (it depends where and when we're talking about) states over the direction of national policy (about slavery) are such a major concern that the political system which develops in the 1820s and 1830s is built to deflect them. The slave states rack up an almost perfect run of the antebellum, from drafting the Constitution to the day before Lincoln's election. Here are some, focused on the more famous:
The Nullification Crisis, as /u/GrandpaZach notes, is a big one. It's also a little hard to classify because unlike the others the South doesn't show a strong sectional solidarity. South Carolina basically went it alone. There was some support in other states, but no one really raced out the gate on a state level to back them. So you can call it a state-vs-nation thing as much as a section-vs-section affair.
Before that, you have the Missouri Controversy. The issue here is whether or not the territory of Missouri, which only has a few slaves, can come into the Union as a slave state or whether Congress can require it to adopt a program of emancipation. This used to be viewed as a tempest in a Congress-sized teapot, of much more interest to the politicians than ordinary people, but recent scholarship appreciates the severity of it more. Missouri got it keep its slaves. To help northerners justify their votes for that, Congress passes an ineffectual slavery ban for most of the rump of the Louisiana territory. Since none of that is organized, or will be, for some time, it's basically a pinky swear that slavery shall not go there. Future compliance not obligatory.
After Nullification, you have Texas Annexation. This is often underrated as a slavery fight because it has a strong partisan element too. (Whigs are opposed, Democrats for.) But the voting does have a sectional lean and opposition to annexation includes that it would add a vast territory which possessed slavery to the Union.
Texas annexation follows right into the Mexican War, which secures to the nation a vast new empire in the Southwest. Antislavery northerners want to ban slavery from that land. So do the people in California, who are part of it and looking for statehood. Southerners are outraged and a multiyear battle ensues over the land's future. The white South wins everything it could expect to win, really, but decides that winning isn't enough and spends the next decade resenting it bitterly and prosecuting that resentment through the new fugitive slave act that many white northerners see as an egregious assault upon their freedom and forcible introduction of slavery into their communities. They're right. The South is overwhelmingly in favor of a leviathan federal government imposing on the states in every likely occasion where they believe that it can further expand and defend slavery.
Things barely settle down at all from that before it's time for the Nebraska Crisis. At issue: whether or not the Missouri Compromise's ban on slavery in most of the Louisiana Purchase will be repealed, thus introducing slavery to a previously free (if colonized by whites) territory. The resolution was that all territorial bans on slavery except for that of the Minnesota territory are repealed in the Kansas-Nebraska Act. This finishes the Whig party and opens up political space for a new antislavery party to become dominant in the North: the Republicans. They elect a president in 1860, the South rebels, and we know the rest.
While it never got to the point of a shooting war the Toledo War between Michigan and Ohio is an example of this. In 1835 the Michigan Territory petitioned for statehood and included a stretch of territory known as the Toledo Strip in its claim. Toledo at the time was an important Great Lakes port, whose control wasn't firmly established as belonging to either jurisdiction due to conflicting interpretations of the Northwest Ordinance, poor survey maps and uncertainty about the extent of Ohio's claim per its constitution.
What ensued was something akin to a cold war between the state and the territory, the former blocking the latter's path to statehood until the dispute was resolved. Militias get raised on both sides with posturing laws on both sides declaring the other outlaw etc. Finally in 1836 a settlement is agreed upon: Michigan will cede the Strip in exchange for the Upper Peninsula. At the time this is considered a raw deal but one preferable to being held in limbo for statehood. Subsequent discoveries of copper, iron and extensive timber in the U.P. slightly offset the losses, and today the incident is known more for being the birth of the Michigan-Ohio rivalry than anything else.
Everyone forgets that the Whiskey Rebellion was the first incident of "conflict " between the states and the federal government. Starting in 1791, the Whiskey Rebellion started when citizens in western PA, in Washington County started to attack and tar and feather tax agents after the federal government instituted a tax on wheat, rye, and corn on farmers needed to pay the debt incurred because of Revolutionary War. The rebellion wasn't a war per se as a series of intimidation actions conducted by citizens against federal employees. Many citizens did take to arms, but President Washington ordered the army into the area until things calmed down.