jmc

We try to solve the equation for a general value of $c.$ If $x$ is an integer, then $\lfloor x\rfloor =\lceil x\rceil =x,$ and so we get the equation $$7x+2x=c,$$so $x=\frac{c}{9}.$ Since $x$ is an integer in this case, this solution is valid if and only if $c$ is a multiple of $9.$

If $x$ is not an integer, then $\lceil x\rceil =\lfloor x\rfloor +1,$ so we get the equation $$7\lfloor x\rfloor +2(\lfloor x\rfloor +1)=c,$$so $\lfloor x\rfloor =\frac{c-2}{9}.$ Since $\lfloor x\rfloor$ must be an integer, this produces valid solutions for $x$ if and only if $c-2$ is a multiple of $9.$

Putting everything together, we see that in the interval $[0,1000],$ there are $112$ multiples of $9$ and $111$ integers which are $2$ more than a multiple of $9,$ for a total of $112+111=\boxed{223}$ possible values of $c.$