Although there are some theories of cross-breeding, humans are a different species to neanderthals.
I don't know where you read that pre-h*mo sapiens averaged over 6ft tall, but it's not true. The fossil evidence from early human race ancestors in Africa clearly show average heights of only 4ft 11in for males and 3ft 5in for females (and that's "stretched out" fully upright since they usually walk stooped).
No doubt there were some taller sub-species just as there are with other animals (although I'd doubt reaching over 6ft). Even today the average height varies from region to region and has also varied at times during history (yes, the war years did mean many children were shorter in Germany), but the general trend is that humans have been getting taller, bigger structured, and bigger brained (although it is diifcult to see what the extra size is being used for in most modern humans!). Some of this is indeed due to medical advances.
In fact, if you're silly enough to go far enough back, the human ancestors were shrew-like / rodent-like animals only a few inches "tall" ... or you could go right back to the primordeal goo and say "human ancestors" were only a few microns "tall". ;-)