Paul Carman, from Addingham, stumbled across the pit, which is legal as long as the traps inside are used to catch vermin and any protected species are released unharmed.
The snare traps in the Nelly Park area of the moorland estate appeared to have been set to target foxes, but the black cat had also been killed.
Mr Carman has taken a slight diversion from the main public access route to avoid a heather burn when he made the discovery on February 12.
However, the estate management defended the use of the traps, claiming that the cat was feral 'with no sign of identification' and could thus be legally controlled as vermin, as the area had a problem with large numbers of cats living wild. They did not clarify whether the animal's body had been taken to a vet to be scanned for a microchip.
It is not thought that anyone has come forward to claim ownership of the cat so far.
North Yorkshire Police have been informed of the incident.
Parts of the Bolton Abbey estate, which is owned by the Duke of Devonshire, are managed for grouse shooting, although shoots were suspended in 2020 and the sport no longer takes place at the Cavendish family's main residence, Chatsworth in Derbyshire.
Large areas of the estate are open to the public as a visitor attraction.
Snares are set on grouse moors to catch foxes that prey on the game birds being reared for shooting. They were found set around a ‘stink pit' - a pile of rotting animal carcasses used as bait to lure foxes.
Mr Carman himself became entangled in one of the snare wires.
“It was extremely distressing to find a dead cat and fox piled up next to snares when fell running on Bolton Abbey estate. As a regular visitor, this is not what I expected to encounter when at the popular beauty spot," he said.
“Bolton Abbey never replied to my email asking about what I found on the estate. They need to acknowledge the strong public opposition to this practice by banning grouse shooting to prevent this from ever happening again.
“North Yorkshire Police did a great job of responding when I reported it to them, but sadly when they arrived to investigate further the cat, fox and snares had been removed.”
The campaign group Ban Bloodsports on Yorkshire's Moors has called for the incident to accelerate a ban on shooting at Bolton Abbey amid a sea change in public opinion.
Last summer, estate director Ben Heyes attributed the decision not to hold shoots during the grouse season to low bird numbers caused by drought and the impact of the heather beetle on habitat quality.
The group's spokesman Luke Steele said: “Visitors to the Bolton Abbey estate will no doubt be shocked to learn that the popular beauty spot is littered with snares and traps, aimed at boosting the numbers of red grouse available for shooting by killing off native predators.
“With grouse shooting put on pause at Bolton Abbey last season, we're urging the Duke of Devonshire to now do the right thing by making this decision permanent. It is clear that the public expects nothing less.”
A Bolton Abbey estate spokesman said: "Much of the Bolton Abbey estate sits within the North Pennine Moors Special Protection Area (SPA). SPAs are selected by central government in conjunction with the Joint Nature Conservation Committee to protect one or more rare, threatened or vulnerable bird species. In this instance, these include the golden plover, hen harrier, merlin and peregrine falcon.
"A key aim of the SPA is to restrict the predation of and disturbance to breeding birds caused by native and non-native predators. To support this aim, we legally control foxes and feral cats within the SPA.
"The snares used on the Bolton Abbey estate are legal and operated by trained staff under a code of best practice, which is endorsed by the Game & Wildlife Conservation Trust. The products used are code compliant and exceed legal requirements. They have a ‘break away’ design that allows larger non-target species to break free. All snares on the estate are set in discreet locations away from public footpaths and private dwellings.
"We work with North Yorkshire Police wildlife officers, who are aware of our predator control in support of the SPA."