Ladybugs can indeed bite. According to Dr. Linda Gilkeson, a former beneficial insect specialist with the Canadian government, ladybugs have no toxins or irritating saliva, so their bites are not dangerous; however, their jaws are just big enough that humans can feel them when they pinch the skin.
Harlequin ladybirds, also known as multicolored Asian lady beetles, have the worst reputation for biting. The bites tend to produce tiny bumps that might sting slightly, though some people are highly allergic. The beetles tend to bite when they are hungry, especially when they have just woken up from hibernation. Because they tend to hibernate in people's homes, humans are often the first flesh they encounter, so they try it to see if it is edible.