virtualbox.org

Steps to reproduce:
1) Create VM1.
2) Create linked copy of VM1 called VM2
3) Rename VM1 to VM1_1.
4) Restart Virtual Box.
You will see that virtual box display error about invalid hard drives images.
VM2 will not start.

I assume you actually mean a linked clone, right?
You can not rename the base image and expect the differencing image to continue to work.
Actually you can't change the name of the *.VDI at all or this will happen on a single install as well.

Moved to Using VirtualBox and this does not constitute a suggestion.