jmc

First, it must be true that $y$ is positive because, otherwise, $\lceil y\rceil +y$ would be negative. Since $\lceil y\rceil$ is an integer, $0.5$ must be the decimal component of $y$. Therefore, $y$ can be rewritten as $x+0.5$. $\lceil y\rceil$ can then also be rewritten as $x+1$. The equation can then be written as $x+1+x+0.5=15.5$. $x$ must then equal $7$, and $y$ can only be $\boxed{7.5}$.