We present a sequence of new linear-time, bounded-space, online bin packing algorithms, the K-Bounded Best Fit algorithms (BBFK), which are based on the ⊙(nlogn) Best Fit algorithm in much the same way as the Next-K Fit algorithms are based on the ⊙(n logn) First Fit algorithm. Unlike those algorithms, whose asymptotic worst-case ratios approach the limiting value of 17/10 from above as K → ∞ but never reach it, these new algorithms have worst-case ratio 17/10 for all K ≥ 2. They also have substantially better average performance than their bounded-space competition, as we have determined based on extensive experimental results, summarized here, for instances with item sizes drawn independently and uniformly from intervals of the form (0,u], 0 < u ≤ 1. Indeed, for each u < 1, it appears that there exists a fixed memory bound K(u) such that BBFK(u) obtains significantly better packings on average than does the First Fit algorithm, even thought the latter requires unbounded storage and has a significantly greater running time. For u = 1, BBFK can still outperform First Fit (and essentially equal Best Fit) if K is allowed to grow slowly. We provide both theoretical and experimental results concerning the growth rates required.