There are no LEGO parts with studs that have no logo on them. The hollow versions of studs which are on technic bricks and many other bricks have the Lego logo printed on the insides.
As someone that goes through a lot of used LEGO, I have found that there are some parts that have been worn and the logo is not nearly as visible as it would be brand new. That said, I have never seen any part be worn to wear the logo has been smoothed out. The ABS plastic that TLG uses for normal system parts is quite hard and would really have to take abuse, not just playwear, to get to a point of smoothness on the studs.
In this individual case, if you look closely at any point in the picture, several parts appear to have smooth studs. This appears to be consistent on many of the parts which would certainly not lead the conclusion of wear. Furthermore, even from the relatively grainy photo, the coloration/shine of the smooth studs parts just seems off...
In any case, it's nearly indefinitely false that the hat pictured is 100% made of LEGO parts.
Upon the added pictures from the creator, it would appear I am wrong. While most of what I said still stands true, I underestimated the power of bad lighting! I've seen some photos that distort the look of LEGO, but never to this extent! I stand corrected.