Each time a cell reproduces the telomeres are cut in half. At a certain pre determined point the cell will cease to replicate itself and die off(apoptosis). As far as I know though cancer cells do not lose telomeres and continue to make complete duplicates including a full set of telomeres.
Yes, telomeres shrink each time a cell reproduces, but only by a bit (not half). Over time, the length of the telomeres are reduced to the point that the new cells are degraded copies of the original - hence the aging process.
The most fascinating news about this discovery, is that scientists have also discovered how to prevent the telomeres from shrinking. In a nutshell, the dream of Ponce DeLeon has finally been realized.