Is there any risk of illness from eating ham?

Yes. Trichinella spiralis, a parasite found on pork, can cause a disease known as trichinosis. The good news is that all processed hams must conform to USDA guidelines to kill trichinae. Caution is required if you purchase an unprocessed ham from a farmer or local market.
 
A  bacteria of particular concern is Clostridium botulinum, better known by the disease it causes: botulism. Canned hams are processed to kill all spoilage bacteria, including Clostridium botulinum. The nitrites used to process cured hams inhibit the growth of Clostridium botulinum.
 
Another bacteria, Staphylococcus aureus (better known as staph) is destroyed by cooking and processing but can be reintroduced to ham as a result of mishandling. The bacteria can then produce a toxin that isn't destroyed by further cooking. Dry curing may or may not destroy staph, but the high salt content on the ham's exterior tends to inhibit these bacteria. Nevertheless, the moister interior allows staph to multiply after the ham is sliced.
 
In addition, mold can grow on hams during curing and drying. Even high salt content and low temperatures don't necessarily inhibit their growth. Most of these molds are harmless. Many are even part and parcel of curing. However, a few can produce toxins capable of migrating into the ham. Moldy sections should be excised from the ham.
 
Source(s):
USDA Fact Sheets "Ham and Food Safety"