Which factions aren't compatible with each other?

[Saphlan]

It's obvious that the imperial legion and the stormcloaks aren't compatible.

But I'm wondering if you can be a part of both the Thieves Guild and the Dark Brotherhood?

 

[KANE OG]

Short answer: Yes

Long answer: Yeeeesssssssss.

They really go hand in hand. The two guilds actually work together often.

I haven't experienced any memorable incompatibilities with anything other than Stormcloaks/Imperials that I can remember, but then again, they all seem to hate me.