Getting sick from expired sour cream is possible but only if the expired sour cream is actually bad and has grown bacteria and mold or has turned bad smelling.
Sour cream can still be good for a week or two weeks past the expiration date as long as the sour cream has remained refrigerated during that time.
The expiration date just means that the sour cream is not fresh anymore and it may not taste as good as before and the texture might change some when the sour cream has reached it's expiration date.
But the sour cream is still safe to eat even when it has expired as long as it's been refrigerated.
If the sour cream tastes bad, smells bad or has any mold growth in the sour cream or the sour cream has become watery looking them throw it out.
Otherwise the expired sour cream will be safe to eat.