I've heard the same thing from doctors; falling asleep in less than 10 minutes is generally considered indicative of sleep issues of one sort or another.

OTOH, ~90% of the population uses caffeine regularly, and given that it takes ~48 hours to completely purge your body of caffeine, I think that saying is only relevant to caffeine drinkers.

Since this is the internet, I am free to reply with anecdata. I get 7.5 hours of sleep just about every night, and I fall asleep in less than 15 minutes more than 90% of the time. I am a coffee addict, but I keep it to 2-3 cups per day (usually). Even the occasional after-dinner cup doesn't usually keep me up.