In this paper, we propose an approach for the nonparametric estimation of the conditional survival function of a time to failure given a time-varying covariate under interval-censoring for the failure time. Our strategy consists in modeling the covariate path with a random effects model, as is done in the degradation and joint longitudinal and survival data modeling literature, then in using a nonparametric estimator of the conditional survival function for time-fixed covariate. We derive the large sample bias and variance of the estimator under simplifying assumptions and we investigate its finite sample efficiency and robustness by simulation. We show how the proposed method can be useful in the early stages of data exploration and model specification by applying it to two real datasets, one on the time to infestation of trees by pine weevil and one on the reliability of a piece of electrical equipment. We conclude by suggesting avenues to make this data exploration method more suitable for formal inferences.