Yes you can get food poisoning from potatoes or any other vegetable if the potato or other vegetable happens to have bacteria on it.
Potatoes and vegetables can have bacteria on them that you cannot see so if the vegetables are old and spoiled and you eat them then they could cause food poisoning or at least make you sick.
If the vegetables such as potatoes or any other vegetables are spoiled then throw them out and do not eat them.
However if you have potatoes that have a few bad spots on them you can cut the bad parts of the potato off and still eat the rest of the potato.
Even fruits can cause food poisoning so never eat bad fruit or bad vegetables.
If the vegetables are just old and do not have any mold on them or they do not smell bad or taste bad then you should not get sick from those old vegetables.
However if the vegetables are really old and moldy or contain bacteria then you could get sick.
Just check the vegetables for any bad smell, bad taste or mold growth and if they are okay then you can safely eat them.
But if the vegetables are old and contain mold, smell sour, or have mold growth then they should be thrown out.
The worst that would happen though if you happen to eat some old bad vegetables is that you get a stomach ache, diarrhea and possibly vomiting.
But just because vegetables are old does not mean they are always bad.
Even canned vegetables can last for 5 to 10 years or longer before they go bad enough to make you sick.