I have most of my planet textures at 4096x4096 playing at 1920x1080 and my system handles everything just fine. Even though the BC engine is old and often buggy, my system specs makes up for the buggy-ness most of the time.
Bottom line, have a good amount of fast RAM installed (minimum 8GB, 16GB or higher MUCH better), a vid card with at least 4GB of video RAM and a decently fast CPU, and the lion's share of the time you'll be fine. That's been my experience of course.
Unless your computer is a literal potato, your system specs are irrelevant when it comes to BC's performance and stability. The game can run fine with even a low end rig. The engine starts to bug out when using more than approximately 1.7-2.0 GB of RAM (even with the 4.0GB aware patch) so anything more than 4GB for your system is redundant. As for GPU memory, the game was designed for 16MB (Mega, not Giga) cards, so even 1GB of VRAM is more than enough.
The 4k planet textures are something the game can easily handle on their own, but the problem is then having any more than about 6 or 8 high quality ship models on top of the 4k planetary textures (and background textures, bridge textures, ect.) leads to massive instability very quickly. And if you are playing the Singleplayer campaign, which loads many different planets for each mission, this problem is even more evident.
Visually, you wont notice any difference between 2k and 4k planet textures unless you fly very close to the surface. It's a small downgrade for a result that uses only 1/4 of the RAM for each planet loaded.