jmc

Let $m=\lfloor x\rfloor .$

If $m\le x<m+\frac{1}{3},$ then $$\lfloor x\rfloor +\lfloor 2x\rfloor +\lfloor 3x\rfloor =m+2m+3m=6m.$$If $m+\frac{1}{3}\le x<m+\frac{1}{2},$ then $$\lfloor x\rfloor +\lfloor 2x\rfloor +\lfloor 3x\rfloor =m+2m+3m+1=6m+1.$$If $m+\frac{1}{2}\le x<m+\frac{2}{3},$ then $$\lfloor x\rfloor +\lfloor 2x\rfloor +\lfloor 3x\rfloor =m+2m+1+3m+1=6m+2.$$If $m+\frac{2}{3}\le x<m+1,$ then $$\lfloor x\rfloor +\lfloor 2x\rfloor +\lfloor 3x\rfloor =m+2m+1+3m+2=6m+3.$$Thus, an integer can be expressed in the from $\lfloor x\rfloor +\lfloor 2x\rfloor +\lfloor 3x\rfloor$ if and only if it is of the form $6m,$ $6m+1,$ $6m+2,$ or $6m+3.$ It is easy to count that in the range $1\le n\le 1000,$ the number of numbers of these forms is 166, 167, 167, 167, respectively, so the total is $166+167+167+167=\boxed{667}.$