Sections 1-5 are concerned with finding lower bounds for the expected sample sizes of sequential multihypothesis tests in the presence of a constraining error matrix. We consider $K$ simple hypotheses corresponding to $K$ density functions $f_i, i = 1, \cdots, K,$ and fix all of the entries of the $K \times K$ error matrix $A = (\alpha_{ij})$, where $\alpha_{ij} = P$ [accepting $f_j \mid f_i$ true]. Lower bounds are found for $E(N \mid f)$, first, when $f$ is one of the $K$ densities, and then, for a $K + 1$st density $f_0$. In Section 6, lower bounds are found when the error constraints arising from the error matrix are relaxed and/or modified. Section 7 finds lower bounds for average sample size when the test is not constrained by an error matrix but rather by a lower bound for the probability of a "correct decision" as a function of the true state of nature. The reader will find that many of the results of this paper extend immediately to a decision theory context with a finite number (not necessarily $K$) of actions or terminal decisions, with $\alpha_{ij}$ denoting the probability of the $j$th action given density $f_i$.