Suckers, to me, refers to shoots growing from the
roots rather than the bark (which I would call watershoots or epicormic growth).
Epicormic growth is an indicator of stress in
trees (and is currently one of the best symptoms to diagnose the
ash dieback epidemic over here).
Assuming that root-suckering can be triggered in the same way, perhaps under-watering, removing leaves or otherwise stressing a tree might force it to sucker. I have also heard that some Prunus species (plum in particular) will sucker heavily when pruned hard which I feel somewhat corroborates this.