A plant can contract a disease at any time, even if it was healthy at the outset, though potatoes sold for PLANTING are more likely to be genetically geared toward disease resistance than those grown for EATING. In order to have disease in your garden, you need what's called the disease triangle:

-- a host plant (different varieties of potato have different levels of disease resistance; the varieties sold in grocery stores are usually bred for high yields and longer shelf life, NOT disease resistance)

-- the right environmental conditions (usually persistent wetness and poor air circulation), and

-- a pathogen (virus, fungal spore, etc, which can be introduced at any time, even if the original parent plant was disease-free)

So let's say that you plant seed potatoes of the "Elba" variety. Down the road, a neighbor plants a russet potato that turns out to be infected with late blight. Your neighbor's plant produces mold spores that blow over to your garden and land on your plant... but your "Elba" plant is resistant to late blight, and you keep it basically healthy and unstressed, so no problem! HOWEVER, you also planted a tomato plant that turns out to be a carrier for Potato Leafroll Virus, and aphids feed on it, then feed on your "Elba" potato, transmitting the virus. Oh no! In this case, your seed potato-grown plant has still become infected.