Can a stochastic process have stationary increments which are not independent increments?
I can see that increments could be independent but not stationary, but can't think of an example going in the other direction, i.e., stationary but dependent.
Let $Z$ be standard normal and consider the process $X_t = tZ$.
There are other, much less trivial, examples (e.g. fractional Brownian motion with $H \neq 1/2$) but this one is fairly immediate to see both properties (stationary but not independent increments)