It does and it doesn't.
While you cannot just grab a cell and change it without also changing the width of the whole column, as you apparently can in Word, the truth is that the Word feature is just a front-end to a split/merge cell engine that makes it look as though a cell is changing independently of the others, but in truth extra columns are being created and cells are being merged to mask it.
You can perform these primitive steps in an InDesign table. It is certainly not as convenient as in Word, but it can be done. You might need to copy and paste cell contents occasionally, too, in order to facilitate the process, making it even more messy.
It is usually easier to make a cell narrower than wider, so when you can, do that. For example, say you want to make the first cell in a row three quarters its current width, thereby making the next cell wider. Do this:
1. Split the first cell vertically.
2. Merge the second half with the old second cell.
3. Grab the border between the first and second cell and move it right so that the first cell is half as wide as it is at this point.
Voila.
If you check the first cell of the row above or below, you'll see that you now have the option of unmerging that cell, even though you never actually explicitly merged it. That's because when you split that individual cell, what InDesign really did was split all of them and then merge all the other rows so that those cells apparently didn't change.
In the table palette, you'll see that the number of columns increased by one when you did this.
But it does get messy as you proceed deeper into the table.
Dave