Place potatoes in medium saucepan; add enough water to cover. Bring to a boil. Reduce heat; cover and simmer 20 minutes or until tender. Drain; cool 20 minutes or until completely cooled.
Meanwhile, place eggs in single layer in saucepan. Add enough cold water to cover by 1 inch. Bring to a boil. Immediately remove from heat; cover and let stand 15 minutes. Drain; rinse with cold water. Place eggs in bowl of ice water; let stand 10 minutes.
Peel eggs; cut into small pieces. Place in large bowl. Peel potatoes; cut into small pieces. Add to eggs in bowl. Add ham, cheese, radishes and onions; mix gently.
In small bowl, combine all dressing ingredients; blend well. Pour dressing over potato mixture; mix well. Serve immediately, or cover and refrigerate until serving time.