Without knowing exactly your setup, I can offer a few items that you might check ...
Do you use a network licensed version?
If Yes Then
Do you have enough free licenses to open LDT2008 (each session requires a different license)
If Yes, then I can't offer any more guidance except to open the drawings in MDI mode (single application multiple documents SDI = 0)
Is your license stand alone? If so, you only have a single license ... LDD may report the license is in use.
FYI ... Whatever version of AutoCAD is the last version to open a drawing, is the default version used to open all drawings by double clicking on the drawing file. This happens because AutoCAD sets the CurVer setting in the registry to the current version running, it could be AutoCAD.Application.15 or AutoCAD.Application.16 ... etc ... and it is referenced in the AutoCAD.Application key .. as such a double click reacts with the AutoCAD.Application key and not the other versions. Thus it could be pointing to the last version used successfully.