I'm apparently not seeing what a "duration" or "interval" is supposed to be such that the answer to this question isn't just obviously Yes. Here's an interval of time: 24 hours. The interval begins when I start the clock running at 12pm Monday, and that interval ends at 12pm Tuesday. It's no good to say "well that's just clock time, not lived, phenomenological time" or something like that, because I experience things beginning and ending and time elapsing in between.
I think that question about immortality is confused. Here's a claim that might be true: I don't experience the end of my life, because by the time my life ends, I'm no longer there to be the subject of that experience. That's an epistemological claim about what kinds of things I can experience happening to me. "I cannot die" is an ontological claim that is not about the limits of my experience. The epistemological claim might be true (although I have my doubts), but the ontological one is clearly false.