No, one thing at a time.
When I was working, I found that I could handle one interruption okay. I would still only do one thing at a time, but I could handle the interruption and then, with ten minutes of mental effort, get back to what I was doing. But if the interruption itself got interrupted, all bets were off. I'd handle the second interruption and maybe get back to the first interruption, but when it came time to get back to the original task, it would take a full half hour to figure out what the heck it had been all about.
In my experience, people who think they are really good at multi-tasking aren't really as good at it as they think they are. They end up doing a poor job of both tasks. It is possible that there are exceptions, but I haven't met one yet.