Do people in the same family suffer from an immune disorder such as allergies, asthma, or arthritis?

It is possible: Allergic diseases can "travel" in families however, parents don't pass on specific allergies or specific diseases. For example, a parent would not pass on their cat allergy to their child and if a parent has allergic rhinitis it does not mean the child necessarily will.

Typically: Tendencies toward allergies and other immunologic diseases definitely run in families, but there's a lot of variation. Allergic parents won't necessarily have allergic kids (70% risk if both parents are allergic), and non allergic parents might have allergic kids. Only an allergic tendency, not necessarily specific allergies, is passed on; kids might not have the same allergies as their parents.