In the case of the DT880, kind of, at least IMO. The way I'd put it is that while not radically different from each other, they're somewhat the antidote for one another. If you feel the HD 600 is a bit too polite and mid-centric, the DT880 has more treble and better bass extension while not having quite as upfront a midrange. Similarly, for those who find the DT880 too strong in the treble, the HD 600 is comfortably more subdued in this range, though with a noticeable bump (especially in direct comparison to the DT880) in the upper midrange/lower treble range that brings forward midrange elements in the mix. Both have tunings that are interpretations of neutral, and while I personally feel the HD 600 is closer (and personally enjoy listening to it more), the DT880 makes a very nice account of itself.
That all said, if you have one of the two and are happy with it, I would say the other isn't necessary unless you're curious how it renders things differently (especially if you find an amazing deal on it).