Identical bikes and a total spoke count

A shop has at least $2$ bikes, all identical; each wheel has at least one spoke, but a bike's front and rear wheels may differ. The shop's total spoke count is between $200$ and $300$. You are told that, if you learned that total, you could deduce the number of bikes for certain. That fact alone lets you work out how many bikes there are. How many?

$17$. The total is $N=b\cdot s$ with $b\ge2$ bikes and $s\ge2$ spokes per bike. The bike count is determined only if $N$ has exactly one such factorisation, which happens only when $N=p^2$ for a prime $p$. The only one in $[200,300]$ is $289=17^2$, so there are $17$ bikes.