In this paper, we study the nonparametric maximum likelihood estimator for an event time distribution function at a point in the current status model with observation times supported on a grid of potentially unknown sparsity and with multiple subjects sharing the same observation time. This is of interest since observation time ties occur frequently with current status data. The grid resolution is specified as $cn^{-γ}$ with $c>0$ being a scaling constant and $γ>0$ regulating the sparsity of the grid relative to $n$, the number of subjects. The asymptotic behavior falls into three cases depending on $γ$: regular Gaussian-type asymptotics obtain for $γ<1/3$, nonstandard cube-root asymptotics prevail when $γ>1/3$ and $γ=1/3$ serves as a boundary at which the transition happens. The limit distribution at the boundary is different from either of the previous cases and converges weakly to those obtained with $γ\in(0,1/3)$ and $γ\in(1/3,\infty)$ as $c$ goes to $\infty$ and 0, respectively. This weak convergence allows us to develop an adaptive procedure to construct confidence intervals for the value of the event time distribution at a point of interest without needing to know or estimate $γ$, which is of enormous advantage from the perspective of inference. A simulation study of the adaptive procedure is presented.