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The number can be anything, sometimes one, sometimes four, sometimes no restriction. I just looked at a set of flights and saw numbers 1, 2, 3, 4 and no limit on the United site.
I think the issue most people can not get is at that instant in time, there may only be one, two, three, or four tickets at that price available. That does not mean they will not instantly move some additional seats into that fare bucket once some of those are sold. If for example you buy two of them, the system may release more instantly so it looks like they were lying.
A better way to look at it, and I just tried this to make sure it was the case.
Take a look at a flight with the warning only four tickets at this price. Change the amount of tickets you want to six instead of one. There's a very good chance you will not be able to get the lower rate, it will price them all at the higher fare bucket.
Example
New York to San Francisco random dates, one ticket, can be had for $278 plus the fees and taxes with a warning that only one ticket is left.
When I change the amount to 2 for the same flights, both tickets now cost $318 each.
Yes there's a good chance as soon as I bought one at $278 another would be made available, but at that point in time, until the computers reprice it all out, there's only one ticket available at that price.
It brings up a good point for people buying multiple tickets (unless there's a reason not to split them). Always price it out as one ticket first, and see if the price changes for adding on more tickets. If it does, book them one at a time if you get the lower rate, or book as many as you can at the lower rate, and the rest at the higher so you don't pay the high rate on all of them. The airline won't ever tell you they can sell you two for one price and two for a lower price (at least online).