The first is a padded vest, the second is padded shoes.
Mushroom Vest (500 gp) This spongy, mottled brown vest is several inches thick and composed of woven strips of preserved fungus. The vest feels soft to the touch and springs back into shape no matter how badly compressed. […] It protects you from impact trauma. When wearing the vest, you treat any fall as if it were 20 feet less when calculating damage.
Boots of the Cat (1000 gp) The boot’s wearer always takes the minimum possible damage from falls (as if the GM had rolled a 1 on each die of damage incurred by the fall) and at the end of a fall always lands on his feet.
RAW there's nothing saying you can't get benefits from both; while the boots specify you automatically land on them, the vest doesn't specify you have to land on it to benefit from it, but does that make any fluff sense? Would you allow it as GM? Perhaps only when the PC rolls out of the fall thus lands first on the feet and then on the vest?
Or could the vest simply be doing something completely magical to protect you from impact trauma? To me it seems the only thing magical about it is that it is incredibly good padding material and regains it's shape well. The last sentence here seems more to mechanically make use of that, based on the assumption that landing on a padded vest (since normally falling and taking damage always means you're landing prone) would hurt a lot less.
Best Answer
Yes.
These items will work together as they affect the fall in different ways.
The Mushroom Vest will reduce the amount of fall damage i.e. the number of dice used to calculate damage taken.
The Boots of the cat affect the amount of damage taken when those dice for falling damage are rolled.
Generally speaking in pathfinder items effects will combine if they affect a roll or situation in different ways or provide bonuses from different sources.
So, by example, for a fall of 60 feet when using both items it would be as if you had fallen 40 feet, 4d6 would be rolled and you would take 4 damage.
You'd be better off with a ring of feather fall (Cost 2200 gp), which would reduce ALL damage for at least [as discussed in another question] 60 ft, for only 700 gp more.
Combining all three items for another example, falling 200 feet.
When falling more than 5 feet the ring will automatically activate and you will float down 60 feet leaving 140 feet to plummet.
The vest then reduces this by 20 feet, leaving 120 feet (12d6 damage)
The boots of the cat would mean this 12d6 delivers 12 damage and you land on your feet (the extra nice thing about the boots of the cat, as taking lethal from falling would normally knock you prone.)
The ring also might re-trigger (see discussion linked), reducing it further. Alternatively crafting it with a higher caster level will make it last for a longer duration.
Fluffwise:
Combining the boots of the cat and the mushroom vest I'd play it as when the player hit the floor they land on the mushroom vest which squashes and absorbs the impact, however it then springs up as the boots of the cat instantly flip them onto their feet, so it looks like they land flat and then immediately spring to their feet in a suitably dramatic pose.