I believe the answer is no. In 1931 Erik Westzynthius showed that there are arbitrarily large gaps in the sequence S(n) of numbers which are relatively prime to the the first n primes. He not only showed that there were gaps of size 2*(p_{n-1}), which would give rise to your conjecture, he also showed that for large n ( greater than something like e^(e^e) ) that the gaps got larger than p_n*f(n) where f(n) is an increasing function involving log(n) and log(log(log(n))), which says that there are long runs of numbers each of which has one of the primes below and including p_n as a factor, and thus gives a negative answer to your question.
It is not known for what n this first occurs (a sequence of length 2*p_n consecutive numbers all of which have a factor among the first n primes). I am working on an upper bound for f(n) currently. For more information, see the related question Erik Westzynthius's cool upper bound argument: update? . I hope to post some information there soon.
Gerhard "Ask Me About System Design" Paseman, 2010.12.14
