Yes. While it is difficult to assess 'real risk' (getting into alt-history territory there), the breakup of the Union was a serious point of concern at least until the ratification of the Constitution. While the American Revolution and the Constitution are often conflated in historical memory, its important to remember that the Constitution was the second attempt at a governing document. The first, The Articles of Confederation was the original plan for the government of the United States. The Articles, which created a government barely resembling the current American government (single legislature, no executive or judicial branches, no power of taxation). Rather than a strong Federal Government, the articles created a system in which the State governments had the majority of the power. Government under the articles lasted ~10 years (created in 1777, ratified in 1781, superseded by the Constitution in 1788) and was fraught with problems. Most famously, the legislative body did not have the power to levy taxes, or collect revenue from the states, they could only 'request' funding. George Washington (along with Hamilton, and others who served in the continental army) were highly critical of the government, both during and after the Revolution. During the war, the legislature was frequently unable to supply funding / supplies for the army as a result of its limited powers. This resulted in Washington's desire for a new government with stronger centralized powers.
When the Constitutional Convention was called (in 1787), its original purpose was to amend the Articles. This was quickly discarded by those in attendance as impractical and they set about creating a totally new system. This is where further threats of breakup occurred, legislators from various states threatened to leave as they didn't think their instructions (to amend the articles) could be expanded to replacing them. During the Convention itself, Northern states threatened secession over the so-called Virginia plan for proportional representation. This threat eventually led a compromise creating the two houses of congress (one according to population, one equal across the states). While this was the most contentious issue during the convention, it was by no means the only one, the Convention was so fraught with disagreement that James Madison remarked afterward that its success was evidence of divine intervention.
Despite the success of the Convention, the threat of disillusion was not over, states still had to ratify it. This fight was also highly contentious, requiring Madison, Hamilton (and Jay) to write the now-famous Federalist Papers in defense of it. North Carolina and Rhode Island (RI hadn't sent delegates to the original convention at all) were the final states to ratify it, in 1790, 2 years into Washington's first term.

Further Reading: Washington, a Life by Ron Chernow (still my favorite recommendation good for a broad sense of the revolutionary era)

Madison, a Life Reconsidered by Lynne Cheney (good for the nitty-gritty details of the Constitutional Convention)

[The Federalist] (https://www.congress.gov/resources/display/content/The+Federalist+Papers)