I have a good job that I really like, but it's not exactly in my field so I keep my eyes open for other interesting opportunities. Recently, a company that is more aligned with my credentials contacted me to set up a phone interview. I often have flexibility during the day, so I told them I was available at the proposed time. Then, a crisis came up at work that day. Leaving for a phone call of any kind would have jeopardized my current job, so I emailed the person who had scheduled the interview to ask if we could reschedule. She obviously didn't get the message because, at the previously scheduled time, I got three phone calls in a row from the company. I was able to stop what I was doing long enough to write another email to apologize, but I didn't get a response about rescheduling. The next day, I got an email that said, "We're sorry we were unable to find a time to talk. Good luck in your job search." Is it really that unheard of to have to reschedule a phone interview? Is there anything I can do as a follow-up to change their minds?
While it sounds like you handled everything absolutely appropriately, I think this serves as a lesson about the nature of the job market. If you couldn't drop everything to talk to these potential employers, they could find someone else who would. There are enough qualified and currently unemployed people looking for jobs that companies don't need to bother rescheduling interviews with people like you.
With candidate searches moving quickly, they may have already made an offer to someone else by now. However, if you really feel like you can't let this go, you could send an email asking them to reconsider another interview time because "I was fulfilling my responsibility to my current employer, and I hope you would want to work with someone who takes their responsibilities that seriously." Sending such an email would certainly be more to appease your own ego than to get a job, though.
I suppose there may also be a way to write something to the effect of, "I'm so sorry that the timing didn't work out for this position, but if your company has other open positions in the future, I hope you'll keep me in mind." Again, I'm not sure this would make any difference. You should just keep scouring job boards. If something else comes up at this company, apply with no mention of this prior misstep.
With this annoying/disappointing situation behind you, I would encourage you to think differently about your job search. Maybe you can't accept interviews during the work day. Or, if you do, you have to take a personal day so that this conflict won't arise again. Maybe next time, you could accept an interview with a caveat that a situation may come up at your current job and ask if they would be amenable to rescheduling if that happens. You could explain that a lot of last minute, unavoidable tasks pop up at your work, and find out how much notice they would need to be able to reschedule.
You say that taking the call would have jeopardized your current job, so I also wonder if it's worth considering whether a prospect is exciting enough to be worth jeopardizing what you already have. If not, weigh whether it's worth applying for in the first place. That sounds pretty drastic, I know, but it may help you prioritize.
Then again, keeping your options open is always a good idea, and protecting your current job definitely sounds responsible, too. So all in all, it sounds like you just happened to encounter a particularly persnickety interviewer. Just as they lost interest in you after seeing that you weren't willing to drop everything for them, perhaps knowing that they have this expectation makes their company less desirable to you as well. In that case, you did well to discover something about the company without wasting any time at all. Good luck with your job search — and I really mean it.