When is a linear combination of independent fBm’s equivalent to a single fBm?

We study and answer the question posed in the title. The answer is derived from some new necessary and sufficient conditions for equivalence of Gaussian processes with stationary increments and recent frequency domain results for the fBm. The result shows in particular precisely in which cases the local almost sure behaviour of a linear combination of independent fBm’s is the same as that of a multiple of a single fBm.