In many biomedical applications, interest focuses on the occurrence of two or more consecutive failure events and the association between event times. Bivariate survival data with interval sampling arise frequently when disease registry or surveillance systems commonly collect data with incidence of disease occurring within a calendar time interval. The initiating event is retrospectively confirmed and subsequent failure event is observed during follow-up. In cancer studies, the initiating and two consecutive failure events could correspond to birth, cancer onset and death. Such data represent a non-randomly screened subset of a population and the interval sampling bias needs to be properly adjusted for in analysis. Similar to truncated survival data, the analysis method for this type of data relies on the key assumption of independence, that is, the disease process does not depend on when the initiating event occurs. This paper proposes a nonparametric test of a relatively weaker but sufficient assumption of quasi-independence based on a coordinatewise  conditional Kendall's tau for bivariate survival data with interval sampling. Further, to quantify dependence between bivariate failure times given quasi-independence, a nonparametric estimator of Kendall's tau that uses inverse probability weights is developed, where the contribution of each comparable and orderable pair is weighted by the inverse of the associated probability. Simulation studies demonstrate that the test procedure and association estimator perform well with moderate sample sizes. The methods are applied to ovarian cancer registry data for illustration.