Situation #1 (emergency button only / Tandem trip)
Resulted in a Hercules aircraft being sent from 500 miles away, it circled above and dropped a radio where I was able to explain the situation which while somewhat serious was definitely not critical. They informed me that a helicopter was being scrambled from 200 miles away. I was able to cancel the emergency heli and request a pick-up by boat. Due to days of bad weather the boat never came, after 5 days a more local heli was able to come and pick us up but we had to leave most our gear and boats behind (months later retrieved by freighter canoe). The whole thing was overkill to the max and VERY costly for taxpayers for what was mostly due to paddler stupidity (I owned a Sat phone at the time, didn't think it would be needed and didn't want to pay the ridiculous cost to get it activated).
Situation #2 (two way communication / solo trip)
A medical situation came up, potentially serious but not immediately life threatening, after a few days of sitting around I realized it was not critical and resumed my trip. After completing the two long portages of the trip but with still another 7 - 10 days to the finish, more big rapids/portages ahead and the late season weather closing in I decided to bail given I could foresee the potential for even greater ugliness. Since I could communicate I was able to arrange a very expensive heli flight out (on my tab) without involving rescue services, had to leave my boat behind but got all my gear out this time.
Last September I returned with a partner to tandem paddle down to my boat and then split up and finish as solos. Boat was where I left it, the trip was completed and all was good.
FYI - After Situation #2 I had to give up some body parts but as you can see they were not parts that would prevent me from continuing to go tripping.