Hi David: theoretically I guess you could, but a problem here is that even though you could add a filter to change the default events slug to, say, get-involved/events it would then be filtered once more by WordPress’s sanitize_title() function and the slash would be removed.
Even so you could probably accomplish this – but you would need to be confident in terms of working with actions and filters (to stop the forward slash from being removed) and also be aware that this would be an unsupported modification – potentially it could cause problems elsewhere that I’m unaware of.