There is a BIG difference between a 14:50 and a 14:15 5k; so for the sake of this discussion; I will assume that you want to run a 14:20. (I could run 14:50 anytime, pretty easily, but never broke 14:20, in my career...(14:22 PR).
I know why I didn't, in retrospect. I didn't do enough fast, track work, necessary to run 12 laps at that pace, comfortably.
The guy who said you need to be able to do workouts like: 5 x 800m around 210-212 and 10 x 400m in 63; is alot closer to the solution than the people telling you you can run that pace by doing 14 mile runs and slower intervals. Yes, that other stuff is all good and necessary, but you need to keep working down to the faster paced intervals, as you are peaking.
Yes, there is always the guy who will tell you that he ran some fantastic times on running 7 minute mile pace, but those guys are far and few between and not the ordinary. I remember the great Amby Burfoot; Boston Marathon winner, top X-C and distance runner of the mid 60s; had a blurb about him in RW magazine, back in the 70's, that said he ran an 8:40 2 mile by only running Long, slow, distance; (LSD), at 7 minutes per mile. Great trade off if you can do it; but 90% of us can't; so don't be misled by 'easy solutions'.