This paper considers the problem of determining a confidence interval for the difference between two treatments in a simplified sequential paired clinical trial, which is analogous to setting an interval for the drift of a random walk subject to a parabolic stopping boundary. Three bootstrap methods of construction are applied: Efron’s accelerated bias-corrected, the DiCiccio-Romano, and the bootstrap-t. The results are compared with a theoretical approximate interval due to Siegmund. Difficulties inherent in the use of these bootstrap methods in a complex situation are illustrated. The DiCiccio-Roman0 method is shown to be the easiest to apply and to work well.