Combine FFT's of shorter length than sample data to get spectrum of all data
Yes you can do it, but you will loose spectral resolution. The method you mention is a variant of the Welch's spectogram estimation algorithm which relies on the block based estimation of power spectrum of a long sequence in terms of shorter blocks within it. 

Given that you have a data of $100$k in length which was sampled at $F_s = 100$k, then a full length DFT after rectangular windowing would provide you a spectral resolution of about $1$ Hz to $3$ Hz, where as a window of $4096$ samples would yield about $25$ Hz to $75$ Hz of spectral resolution.

If this resolution is enough for your purposes, then you can (and indeed you should better) perform the block based averaged spectral analysis in order to trade-off between the reduced spectral resolution and an increased estimation accuracy (decreased estimation variance)