Why do certain signals in proton NMR experience extensive splitting despite only having 1 neighbour?
This question has a few different parts to it.

Part 1 - what's that peak at 3.6ppm?
Well, it's not the -OH peak.  The peak at 3.6 is from the CH(OH). 

Part 2 - where is the -OH peak if that isn't it at 3.6ppm?
Peaks from labile protons are frequently not observed, and have variable chemical shifts, unless great care is taken in sample preparation. This sample was run in CDCl3, and the -OH peak exchanges with H2O, and comes at ~1.6ppm.

Part 3 - can we account for the splitting of the peak at 3.6ppm now we know what it is?

The CH(OH) proton should couple to proton on the adjacent carbon (split to a doublet) and also the protons of the methyl group (split to a quartet). It won't couple to the -OH for the same reason we don't readily observe the -OH; it is rapidly exchanging with water. So, we expect that the peak at 3.6ppm should be a doublet of quartets. Looks more complicated than that to me.

Part 4 - It looks  more complicated that a doublet of quartets to me. What's happening there? 
Of course, 3-methylpentan-2-ol has 2 stereocentres, and so we have here a mixture of diastereomers in solution in the sample. Both diastereotopic CH(OH) protons have very similar chemical shifts, and so there are in fact 2 sets of doublets or quartets. It's also why the rest of the spectrum looks such a mess.