Stochastic processes and Markov chains

Question

In my course i have defined stopping times, first passage times and hitting times, where the first passage time is $T_{i,j} = \inf( n>0 : X _n=j)$ and $H_i=\inf(n\geq0 : X_n=i)$. stopping times are those that only look into the history/ past of the Markov chains to see if the event has happened. However, we had also defined excursion times, which was $S_i^r= T_i^r-T_i^{r-1}$, which we deduced is not a stopping time. I don't understand why this is the case since $S_i^r$ only looks into the past to see if the events have happened, and by the SMP, even though both $T_i^r-T_i^{r-1}$ are random stopping times, we can condition on them.