You can install some other Linux distro on the partitions holding the distro that you want to get rid of. Be sure to allow installing only on those partition(s). Try to use the same partitioning scheme, if you have separate /, /usr, /var etc partitions.
During installation avoid/skip the step where the GRUB settings are supposed to be written. You then have to adjust your GRUB configuration from the other Linux distro or from a Live-CD by hand to include/adjust the new Linux system. Or boot from the GRUB command-line directly into the newly installed system (you need to read the GRUB manuals before attempting this).
Depending on the distro that you want to install, the installer could have an option to create a new GRUB setup including all the other operating system - this would be the easiest way to deal with the situation (should work for Linuxes, but Windows is always tricky).