Yes but …..
As an English Country Dancer - i.e. a country dancer in England, I dance what we call 'Playford', 'Ceilidh' and 'American'
Certainly Ceilidh dances are usually published with suggestions for tunes '32 bar reels' '48 bar jigs or reels' etc. And each band has a repertoire of tunes that they play well. If they play regularly for a particular caller tunes and dances that match well became paired.

I think American 'English Country Dancers' dance what we would call 'Playford'. For that tunes and dances have become linked. The tunes may be a less standard length. The dancers may 'know' the dance by responding to the music. Even in Ceilidh this can sometime be true - easy to go wrong in Blaydon Races if the band have gone on to a second tune and I can imagine getting very confused if I tried dancing a different dance to the tune Blaydon Races.

Historically there was more mix & match, by publishers. So new sets of figures were put to old tunes and in different books you find the same figures to different tunes (and the dance might be called after the tune). Cecil Sharp searched out old Playford dances from the 17th century, but if he didn't like the tune Playford had matched it with he used another from the book e.g. Hit & Miss to the tune Daphne.
So yes, you can, but the dancers may not like it if they already know the dance. What some bands in England do is find a second tune that fits with the first and play a medley. This would give the dancers and caller a chance to see if it suited the dance.