This is very typical in my experience. I have shot 20 elk or so, and been present when more than twice that many were shot. It didn't seem to matter much what they were hit with, from 7mm-08 to .375H&H, the "ten seconds to die" rule is pretty accurate for a double lung shot. DO NOT expect an elk to fall over at the shot. it might happen, for some odd reason some of the quickest kills I've seen have been with little bullets from .270's and 7mm's, but you can't count on seeing a reaction or getting instant results on any thing but a shot to the spine, brain, or both shoulders.
The advice I'd give you is instead of wondering if you'd missed, shoot again! Far too many elk are hit and lost by hunters who "admire the shot" and fail to reload and shoot again. I've shot quite a few elk unessesarily a second or third time. But it sure as heck beats a long frustrating tracking job.
