I could see several possible historical points for Balkanization.
1) Right after the revolution. If they hadn't been able to reach agreement, the states could have dissolved the Articles of Confederation and gone their separate ways. Had this happened, I could see perhaps 3 "countries" -- 1 of most of the NE colonies/states, 1 of some of the southern colonies, and perhaps 1 state choosing to join neither. In any case, they might eventually have joined together (in some form) anyway.
2) Civil War, obviously. Eventually a stalemate could have been reached, leaving two countries. However, with less centralized power (in either), other Western States might potentially have emerged as independent nations. For exampleTexas leaving the Confederacy, seizing land in New Mexico or Oklahoma territory, and being independent.
3) Future. While there's occaional talk of states seceding, it won't go anywhere in the near term because cooler heads will prevail and they'll conclude (as Scotland did) that it makes more economic sense to say. However, if over the course of a few decades the US had a disasterous major war, an economic collapse, or a cataclysmic natural disaster, then some states might break off in the chaos and the remainder might be unwilling to fight to pull them back in.
What do you define as "balkanization"? I am partial to immediately post-revolution division, ending up with a New England, a NY-PA-NJ republic, Virginia (possibly including NC, Kentucky, and Tennessee), and a South Carolina-GA state. Maryland can be Belgium. What happens next is up in the air.
In the Civil War, depending how it starts, there could be a New England, independent or in union with the rest of the North, and an independent California and Texas seem likely. I doubt the CSA would be able to stay together for long, so we might see a division between SC and VA (that is, if the USA is too busy to take them back).
Also, don't forget the Oregon Territory. I can see it becoming a British Dominion separate from Canada if the US stays out.
I think any U.S balkanization is doubtful after the civil war. Before that, it was called These United States of America. Afterwards, it was called The United States of America. This change is for a reason: America was more centralized and had an even stronger identity, with state-identity fading into the background.