If only I had a yen for every time this question came up in the Fujifilm SLR/X series forum at DPReview, I'd... be able to buy a chocolate bar or two.
There are some good answers in the thread for anyone that looks for community consensus.
My short answer: if you are shooting anything remotely action-oriented outdoors in bright sunlight, you will not be able to see your subject. Even today's best EVFs lack the contrast and brightness necessary to stand against the sun, not to mention refresh rates that are able to obviate vertigo.
If you are in a cold environment and need to change focus parameters, among other things that rely on the back panel, the X-T1 won't cut it. It is unbelievably clumsy to use with gloves on. The rear AF button isn't the do-all AF-ON button that it is on Canon/Nikon bodies.
I am ecstatic about the X-T1. It shows Fujifilm improving on most things. But the X-T1 is in no way a replacement for a DSLR in the arenas where DSLRs reign supreme. A great mirrorless camera is still a not equal to a great DSLR of any stripe unless weight, size, lens selection (this refers to APS-C DSLRs), are the deciding factors.
If you are a still life or street shooter, or work with families and portraiture, the X-T1 may do what you need it to do. If you use manual lenses and love shooting slow subjects wide-open, the X-T1 is great. But there are many ifs/ands/buts left. It is far from being perfect. I am not quite ready to report my full findings.
In many ways, choosing a mirrorless (or any camera for that matter) is like choosing a lighting setup. I've done it like this: Elinchrome for sports, Profoto for studio, Quadra for events where small lighting setups are necessary. No one light company can fulfil all your needs. Ditto cameras. The X-T1 has a niche and it fulfils its purpose in that niche. It is NOT the everycamera.
My hope is that Fujifilm will continue to develop imaging to cover necessary, and specific niches and try not to tackle everything under one bonnet. I wish that they continue to do a job well and do it with their own colour.