I briefly owned a Fuji X series camera and got the impression that while it had less noise at high ISOs, this was merely because it applied aggressive noise reduction even to RAW files - its files had much less detail than equivalents from my other cameras, mirrorless and otherwise (in fact I noticed a relative lack of sharpness at any ISO, an impression confirmed by just about every photo I've seen online taken with any Fuji X-series camera). For that and other reasons I returned it. I really wanted to like the images it created because I find the cameras themselves very appealing aesthetically and in terms of design, but among mirrorless cameras I prefer the results from my Olympus and Sonys. This seems to be rather a minority view, however.
Not sure what you were shooting but I've got OOC jpg output from an XM1 and kit lens that is terrifically sharp even at 6400. Xt1 and long cheap kit lens similarly impressively sharp. Posted one here a while back.
perhaps you were suffering blur from some non-stabilized lenses?
I used a XE1 + the much-admired kit lens, plus, via adapters, several MF lenses. What I was seeing wasn't motion blur, and close supbjects photographed better than more distant ones. It could, of course, be that the lens or body were defective in some way (the images I liked most were taken indoors, hand held, in very low light with MF lenses). That said, I've looked at a lot of images online posted by admirers of Fuji X cameras and lenses to demonstrate sharpness etc. and none of them struck me as very sharp (not bad, but nothing that seems to me to justify all the claims made for them, especially given how much the equipment costs); that at least one professional photographer (soundimageplus), who claims that Fujis are his favorite cameras to use, concedes that the resulting images aren't as good as he gets from his Sony A7/r (though it's hard to tell from his photos - the images he posts are too small and I don't like the way he processes his photos); and that the comparative tool at dpreview, for whatever that's worth, confirms my impressions.
But I'm certainly willing to try one again - maybe I'll rent the latest one, with a prime or two. If your rabbit photo is ISO 6400, that's not bad at all.... Even so, they seem very expensive for what they are; for the same price as a Fuji Xt1 you could buy two Sony a6000s or almost three Canon SL1s; or for a mere $200 more you could move up to a FF Sony a7r.