I can’t drag parts around unless I mouse over an area above about -100, in which case I can then drag it anywhere as expected. Move and Rotate tools work normally.
This happens consistently while building caves underneath my map (part in gif is at 0, -100, 0 on a squished baseplate):
(My FallenPartsDestroyHeight is -50000)
It isn’t always at exactly -100
edit: Replaced dead gyazo link with mp4
Are you sure it is related to the height you are building at? I’ve been trying it out and it seems when you try to drag a part at that angle against any wall it is resistant to being dragged downwards initially.
Have you tried grabbing it on the end closer to the wall, rather than the outside? Usually, when grabbing parts, it sometimes takes a few pulls to ‘un stick it’ from its position depending on the way I grab it, or at least I’ve found.
My mushroom placeholder wont move 
and after translating the map by (0, 1000, 0):
I recently added the water so it’s not that.
I’d build higher but I gotta get all that juicy precision
Edit: Fixed dead gyazo links
I have had this issue too when building at negative Y coordinates. Can confirm it consistently happens, and a fix would be greatly appreciated.
I’ve also been doing underground building recently and I can confirm that this does happen. It can happen much higher than -100; I’ve had it happen around -24 and -74.
Can someone attach a place so I can take a look?
Thanks! That helps a lot! It looks like it does drag correctly below -100, but the parts are “sticky” for some reason, like ZacAttackk mentioned. I’ll add it to our list of things!
Just noticed you can drag parts across a plane that is located at -25 for me. Probably related
It might be determined by the cframe of the camera relative to that plane
I’ve noticed another thing that might be related to this. If you copy and paste a part that’s below y=0, the pasted part will be placed above the y=0 plane even if there’s enough space above the original part.
This is a different issue. It also happens when creating a part. There is a fix for it coming soon.
We fixed this last year! Sorry it took so long to confirm, and a huge thanks for the report.