Can cats become ill if they eat wet cat food that's been out for a bit?

Of course. Imagine yourself eating a piece of stale food or something. Cats shouldn't - by rights - eat decying food unless they are starving. Dry food goes stale quickly - not just in the bowl (be sure to change the food every day) but in the packaging, too. Bacteria can grow and multiply in the kibble which can - and has - cause serious health issues to cats.

If it's a well fed and healthy cat, it should turn it's nose up at the decaying and stale food. Be sure that your cat gets fresh food; don't risk your cat getting ill!

I had just bought some dry food last week and I lost my older cat of 19 a few days after I had opened the food and I am now noticing my other cat who is 5 doing the same thing as my 19 yr old did before he died. I have since stopped giving him the food and have already noticed a improvement in his health.