If $[x]^2-5[x]+6=0$, where $[\cdot]$ denotes the greatest integer function, then

Given $[x]^2 - 5[x] + 6 = 0$, let $y = [x]$ to obtain $y^2 - 5y + 6 = 0$. Factoring yields $(y-2)(y-3) = 0$, so $y = 2$ or $y = 3$.

Since $y = [x]$, the solution splits into two cases: either $[x] = 2$ or $[x] = 3$. By definition of the greatest integer function, $[x] = 2$ implies $2 \le x < 3$, while $[x] = 3$ implies $3 \le x < 4$.

The union of these intervals is $2 \le x < 4$. Comparing with the options, $x \in[2,4)$ matches this solution.

Final answer: $\boxed{\text{C}}$