In order to increase the storage limit for upgrade kits (there are 5000 slots per kit category by default), you need to increase your VIP level.

VIP5 adds +500 slots for upgrade kits
VIP6 +1500
VIP7 +2500
VIP8 +3000
VIP9 +4000
VIP10 +5000
VIP11-12 +10000
VIP13-15 +15000