Given a set P of words, the Shortest Linear Superstring (SLS) problem is an optimization problem that asks for a superstring of $P$ of minimal length. SLS has applications in data compression, where a superstring is a compact representation of $P$, and in bioinformatics where it models the first step of genome assembly. Unfortunately SLS is hard to solve (NP-hard) and to closely approximate (MAX-SNP-hard). If numerous polynomial time approximation algorithms have been devised, few articles report on their practical performance. We lack knowledge about how closely an approximate superstring can be from an optimal one in practice. Here, we exhibit a linear time algorithm that reports an upper and a lower bound on the length of an optimal superstring. The upper bound is the length of a superstring. This algorithm can be used to evaluate beforehand whether one can get an approximate superstring whose length is close to the optimum for a given instance. Experimental results suggest that its approximation performance is orders of magnitude better than previously reported practical values. Moreover, the proposed algorithm remain efficient even on large instances and can serve to explore in practice the approximability of SLS.