It is not that you can't measure less than 1cm (10mm) it depends on the accuracy you are after.
If you have a mm scale the measurements could be out by +/- 1mm and for something less than 10mm this is worse than a +/-10% error. In most physics experiments that would be considered not good enough.
A vernier scale gives you 1 more significant figure so the error could be +/- 0.1mm.
This is better but as you get less than 10mm it may still not be good enough and you need to resort to another measuring instrument... micrometer, traveling microscope etc.