During the early part of the Covid-19 pandemic, national and local governments introduced a number of policies to combat the spread of Covid-19. In this paper, we propose a new approach to bound the effects of such early-pandemic policies on Covid-19 cases and other outcomes while dealing with complications arising from (i) limited availability of Covid-19 tests, (ii) differential availability of Covid-19 tests across locations, and (iii) eligibility requirements for individuals to be tested. We use our approach study the effects of Tennessee's expansion of Covid-19 testing early in the pandemic and find that the policy decreased Covid-19 cases.