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We use the conformal bootstrap to study conformal field theories with $O(N)$ global symmetry in $d=5$ and $d=5.95$ space-time dimensions that have a scalar operator ${\ensuremath{\phi}}_{i}$ transforming as an $O(N)$ vector. The crossing symmetry of the four-point function of this $O(N)$ vector operator, along with unitarity assumptions, determines constraints on the scaling dimensions of conformal primary operators in the ${\ensuremath{\phi}}_{i}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\phi}}_{j}$ operator product expansion\ifmmode \dot{I}\else \.{I}\fi{}mposing a lower bound on the second smallest scaling dimension of such an $O(N)$-singlet conformal primary, and varying the scaling dimension of the lowest one, we obtain an allowed region that exhibits a kink located very close to the interacting $O(N)$-symmetric conformal field theory conjectured to exist recently by Fei, Giombi, and Klebanov. Under reasonable assumptions on the dimension of the second lowest $O(N)$ singlet in the ${\ensuremath{\phi}}_{i}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\phi}}_{j}$ operator product expansion, we observe that this kink disappears in $d=5$ for small enough $N$, suggesting that in this case an interacting $O(N)$ conformal field theory may cease to exist for $N$ below a certain critical value.