If you consider that you could go out on a seven-day trip, get hurt on day 2, and no-one would report you missing until five days later, then obviously it's foreseeable that you could be out more than three days.
Once you're reported missing, though, if you left a good trip plan and stuck to it you should get found within a day.
And it bears pointing out that if you take "prepare for the worst" seriously, if you accept that serious injury can happen to you, and you travel solo in the backcountry, then you should worry less about surviving seven days and more about where to buy a PLB or SPOT.