I like Aaron Meyerowitz's efforts and think his and similar methods deserve further study. I want to post my skepticism as a counter, and hope that something will arise from the contrast. I do not consider this post as being an acceptable answer though.
The problem is essentially a shifted sieving problem. After taking the first $n$-many (finitely) primes $q_i$ with offsets $r_i$, one has an eventually periodic pattern of uncovered integers which repeats with period $Q_n = \prod_{i \leq n} q_i$, which contains $U_n = \prod_{i \leq n} (q_i - 1)$ uncovered numbers in each period, and has the first period starting somewhere near $M_n = \max_{i \leq n} r_i$.
If the $q_i$ are the primes in ascending order, we have (Mertens) that $U_n$ is $O(Q_n/\log(q_n))$, which is (roughly) about $n$ times as many primes in the interval $(M, M + Q_n)$ when n gets large, especially when $n$ is comparable to the largest integer allowed to be uncovered.
If the distribution of coprimes to $Q_n$ were amenable to being nicely covered by arithmetic progressions of primes less than $q_n$, I might share Aaron's confidence. However, each later prime $q$ used is itself coprime to $Q_n$, and with small deviation will cover only about $1/q$ of what needs to be covered. I suspect that when $n$ gets to be about $Q_{24}/2$ using Aaron's sequence $Q_i$, he will run short on primes. It might be prudent to try more extensive simulations which leave no numbers greater than 50 uncovered.
Gerhard "Saying As I Feel It" Paseman, 2011.11.18