Absolutely - if you know what you're doing.
Mounted games riders mount and dismount at speeds up to the canter all the time. It's often called vaulting (not to be confused with the discipline of "vaulting"). Many horsemen learn the emergency dismount.
An emergency dismount means you generally let go of the horse (it is never safe to try and keep hold of a horse when falling off). A speed dismount should have you land facing the same direction of the horse and running - which is how I ended up in the ER. I forgot the horse I was riding was four inches shorter than the one I normally rode...couldn't walk for two days.
But this is definitely something you can have a character do if they know how to ride. (If they don't, then they are likely to get hurt and even if they do, it can be risky, but sometimes staying on is far more dangerous). And yes, it's also possible to get back on at speed.
Which leads me to the next thing I plan on talking about...the cowboy "jump into the saddle from an upper window" trick and how feasible it is.