Confirmed. Looks like a problem with the Layer Properties dialog because changing that option via Maxscript works.
Make sure you report this - Help > Speak Back > Report a Problem, or go
here (same place either way).
Workarounds.
If you have access to any previous releases (2014/2013/2012) save the scene as the appropriate version, open in the older version and change the Display setting as required and save the scene (under a new name, just in case). Open in 2015 and the setting will now be "stuck" on the one you chose.
Alternatively it can be changed by a short Maxscript.
Replace the "layerNumber" with the number of the layer (as shown in the Layer Explorer) noting that the top one is layer 0, the second is 1 and so on
when they are sorted by Name, ascending. Replace displayType with one of #viewport, #boundingbox, #wireframe or #shaded.
(layerManager.getLayer layerNumber).display = displayType
So, for example:-
(layerManager.getLayer 4).display = #shaded
Max 2016 (SP1/EXT1)
Win7Pro x64 (SP1). i5-3570K @ 4.4GHz, 8Gb Ram, DX11.
nVidia GTX760 (2GB) (Driver 430.86).