Basically the rule is: If you miss a connection due to your own fault, you cannot just hop on another train. However if you miss a connection due to a fault of a transport company, you can just continue on the next available transport. And that even applies at the beginning of the trip.
I have been in this situation a few times. Once I was travelling Wengen - Rolle, and had a supersaver ticket from Interlaken (as my Jungfrau Summer Pass is valid till Interlaken). The BOB trains, which is normally very reliable, that day had to stop, because a heard of cows was blocking the tracks. All very entertaining and hilarious, but we missed our train in Interlaken. No problem. Just took the next train, and the conductor had no issue with it.
Another case was when I returned from Vienna on the NJ, and had a supersaver for Zurich - Interlaken. NJ was delayed (As sometimes happens) and I used the ticket on the next train to Interlaken without any issue.