Yes, cats can get bitten by brown recluses spiders in the USA where this poisonous spider is found (see map). In fact, domestic and feral cats are (1) curious and excellent predators and (2) insects are on their list of prey items. This means that they are at risk of being bitten by small, poisonous creatures such as the brown recluse spider.
The danger for cats is to be bitten on the face and/or paw. How many times have you seen a domestic cat curiously pushing their nose up to a spider to check it out in all innocence before eventually killing and eating it. As the brown recluse carries venom the sting is toxic to animals although the spider is not inherently aggressive. But being crudely pushed around by a cat will make it retaliate.
The cat feels a sharp pain when bitten by the spider. This may be followed by these symptoms: laboured breathing, fever, excitability and chills. There may be seizures.
The brown recluse venom causes blistering of the skin with accompanying pain. There may be a bull’s eye lesion (damage to the skin). After 7-14 days the area may ulcerate and the skin dies. The wound will heal slowly due to its size and severity.
A second condition may develop causing fever, joint pain, possible vomiting and seizures. There may be kidney failure and blood disorders. These symptoms are fairly rare but frequently fatal.
There is an antivenin. A rapid visit to your veterinarian is called for.
Note: Glue traps are a good way, I am told, to capture brown recluse spiders and to find out if you have them in the home. They can be purchased online or in hardware and grocery stores. For spiders the trap should not have a raised edge. Use as many as you can tolerate. They should be placed in corners and along edges where wall meets floor and where there is clutter. Bites from this species of spider are rare even where they are abundant.