jmc

First, we note that $r$ must be positive, since otherwise $\lfloor r\rfloor +r$ is nonpositive. Next, we know that the decimal part of $r$ must be $0.5$. We write $r$ as $n+0.5$, where $n$ is the greatest integer less than $r.$ Therefore, we can write $\lfloor r\rfloor +r$ as $n+n+0.5=15.5$. Solving, we get $n=7.5$. This is impossible because $n$ must be an integer. Therefore, there are $\boxed{0}$ values of $r$ such that $\lfloor r\rfloor +r=15.5$.