What sort of grid are you using and what sort of clearance rules? I've had issues when using imperial trace/space settings and metric grid and traces placed in "hug obstacles" mode. I suspect it's some sort of rounding error that causes Altium to think that there's a DRC violation when you're trying to move a trace, even though DRC says it's okay (because it actually is). On the same project I also had problems when moving a group of traces all at once suddenly had DRC violations, even though the spacing between them shouldn't have changed at all.
Could also be a problem with snap settings, where the next place the tracks want to snap to would cause a DRC violation, so Altium won't let them go there, leaving them stuck in place. Maybe try a smaller snap grid.
One more basic thing, is the layer the tracks are on active?