Which filaments actually do need to be stored in a drybox?
PVA
From experience I can tell that PVA filaments need to be stored with silica beads in a plastic bag or in a specific dry-box. PVA is soluble in water and is very hygroscopic. With moisture it gets soft and swells. My Ultimaker came with an open spool of PVA which popped when heated (steam bubbles popping) resulting in very poor quality supports and clogging of the nozzle. A newly bought spool which was properly packed did not have these issues.
Nylon
Some Nylon filament brands require to be stored dry or need to be dried before printing. I have a spool of Nylon that has taken up some moisture although carefully packed, the only time it was out is when it was being printed. This experience applies to Ultimaker Nylon; I've learned now that not all Nylon filaments are behaving the same, it is suggested to look-up the specifics from the manufacturer or from reviews prior to buying.