This is because if Americans can remember the national holidays of two countries, they're the U.S. (July 4) and Canada (July 1); if they can remember a third there's a good chance it's France (July 14).
What are the chances that two countries which border each other have their national days within three days of each other? You can pick when the first one should be at random, on the Nth day of the year; then the other one must have its day sometime between N-3 and N+3, a seven-day span So the probability is one in fifty-two; one expects there to be two countries which border each other and have national days within three days of each other.
Two bordering countries have a one-in-365 chance of having the same national day, if such days are chosen at random. But since such days often commemorate historical events, and two adjacent countries probably share some history, I'd think that such events are more likely than one in 365.
Of course, this is all made a bit trickier by the fact that some countries seem to have more than one "national day". Mexico's Independence Day, for example, is September 16 -- but I bet a lot of Americans thinks it's May 5. Canada celebrates July 1 -- but Quebec calls June 24 "la fête nationale du Québec".
A quick look at the Wikipedia lisf of national holiays reveals the following coincidences:
- Canada has Canada Day on July 1; the U. S. has Independence Day on July 4
- Pakistan has Independence Day on August 14; India has Independence Day on August 15
- Costa Rica, El Salvador, Guatemala, Honduras, and Nicaragaua all have Independence Day on September 15; Mexico has it on September 16. (Chile declared its independence from Spain two days after Mexico did, but it looks like they were separate events. Belize declared Independence on September 21, but over a century later.)
The bunch of Central American countries on September 15 all commemorate the same event, the formation of the Federal Republic of Central America in 1821, which split up into those countries about twenty years later. Mexico declared its independence on September 16, 1810.
The Canadian and American holidays of course celebrate different events -- unless somehow Canada started in Philadelphia and nobody told me. (Incidentally, the Continental Congress actually voted for independence on July 2. On this basis I think I should not have had to serve jury duty on Monday, since that should have been a holiday -- and less than a mile from where it all happened, no less!
The Pakistan and India days seem to both commemorate the 1947 partition of India; it's not clear to me why they're not the same day.
But there seem to be at least two cases in which adjacent countries celebrate their national days within three days of each other and they're not commemorating the same event -- namely Mexico-Guatemala and U.S.-Canada. (And, of course, the U.S. borders Mexico.) This is less than I would have expected -- you'd expect two such near-collisions if there were about 104 borders between countries in the world, when there are clearly more than that. But I am working from a list which is clearly incomplete. This list is more complete but not sorted by date.