The easiest-to-visualize example I've come across is geodesics ('straight lines') on the surface of a sphere; all geodesics satisfy "minimal action", but they're only guaranteed to be an actual minimum if they're "sufficiently short". For example, there are two geodesics connecting Los Angeles, CA to New York, NY on the surface of the Earth -- one going across mainland USA, and the other going around the other side of the world. Both are "minimal" in the sense of an action principle (by definition -- they're "straight lines" / geodesics), but obviously only one is a truly minimal path. Now if you were restricted to choosing paths that were shorter than one half the circumference of the Earth, you would be guaranteed to have a minimum -- due to peculiarities of a sphere.
Hopefully I got all of the facts right there, and it was clear enough to at least get the general idea across...if I failed on either front, hopefully someone will correct the record.