I was reading this question when I spotted something interesting. "The Sound of Her Voice" opens like this:
LISA [OC]: My name's Lisa Cusak. Until a couple of days ago, I was the commanding officer of the Olympia.
SISKO: The Olympia.
LISA [OC]: We left the Federation over eight years ago for a long range exploration of the Beta Quadrant.
SISKO: What happened to your ship, Captain?
LISA [OC]: We were finally heading home, if you can believe that, then we picked up some strange energy readings in a nearby star system, and I decided to stop and investigate. We found an energy barrier around the fourth planet that was unlike anything we'd ever seen, and when we probed it with our scanners it triggered a quantum reaction. There was an enormous surge of metrion radiation that disabled our engines. The next thing I knew, we were spiraling in toward the surface. I gave the order to abandon ship and the last thing I remember is a console exploding in my face. I woke up in an escape pod on the surface and I've spent the last day and a half sitting in this cave trying to raise someone on subspace.
BASHIR: Captain, Doctor Bashir, Chief Medical Officer. Your message said that you were on a L class planet. Are you sure?
LISA [OC]: Positive. And to answer your next question, yes, I've been giving myself fifteen cc's of triox every four hours to compensate for the excess carbon dioxide in the atmosphere. Just like it says in my medical tricorder.
BASHIR: How much triox do you have left?
LISA [OC]: One hundred and fifty millilitres.
BASHIR: Will you to decrease the dosage, Captain, to eight cc's every six hours. We need to stretch your supply as long as possible.
KASIDY: What happens when she runs out of the drug?
LISA [OC]: That's a good question, Doctor. What happens then?
BASHIR: You will begin to experience the effects of hypoxia. But before that happens, the triox compound will have strengthened your cardiopulmonary system, allowing you to better withstand the effects.
LISA [OC]: Better withstand the effects. In other words, I'm going to be gasping for air and turning different shades of blue by the time you get here.
BASHIR: Yes, I'm afraid so.
LISA [OC]: Thanks for brightening my day.
KASIDY: Is there anything we can do?
LISA [OC]: There is, actually. I can't sleep. I think the injections are keeping me awake and I haven't had anyone to talk to for two days.
SISKO: We'll be able to help you with that, Captain. I'll have one of my officers stay on the comm. line with you at all times.
Star Trek: Deep Space Nine episode 6x25 "The Sound of Her Voice"
At the end of the episode, it is revealed that:
The Olympia had crashed more than three years before the Defiant arrived, and the planet's energy field shifted their radio signals back and forth through time. Lisa was long dead when they finally arrived on the surface.
She told them that she left the Federation "over eight years ago," but actual records would have reflected 11+ years. The Olympia was heading home at the time of the crash, so it is likely that Starfleet would have declared them overdue at some point, meaning 8 vs. 11 years is not a minor difference. It's the difference between being overdue and not.
She also specifically told them that she had only been on the planet surface for "a couple of days," so it is not possible that Sisko et al. misinterpreted the "eight years" as the mission duration rather than the total time elapsed. Such a misinterpretation would require her to have been stranded for more than three years, not two days, which is inconsistent with what she told them.
Why didn't anyone check Lisa's story against Federation records and note the discrepancy? At the very least, it seems like they would have contacted Starfleet to say "Hey, we're rescuing this person, if we don't report in by X time, assume we got stranded too and send another rescue crew."