Why does my TreeSet not add anything beyond the first element?

If you put your objects in a TreeSet, you need to either provide an implementation of the Comparator interface in the constructor, or you need your objects to be of a class that implements Comparable.

You said you implement compareTo from the Comparable interface, but in your comment you say that you didn’t, so am I correct in assuming that you just return 0; in the compareTo method? That would explain your problem, because TreeSet would then think that all your objects are ‘the same’ based on the compareTo method result.

Basically, in a TreeSet, your objects are maintained in a sorted order, and the sorting is determined by the outcome of the Comparable/Comparator method. This is used to quickly find duplicates in a TreeSet and has the added benefit that when you iterate over the TreeSet, you get the results in sorted order.

The Javadoc of TreeSet says:

Note that the ordering maintained by a set (whether or not an explicit
comparator is provided) must be consistent with equals if it is
to correctly implement the Set interface.

The easiest way to achieve that is to let your equals method call the compareTo method and check if the result is 0.

Given your PatientImpl class, I assume that you would want to sort patients first by their last name, then by their first name, and then by the rest of the fields in the class.

You could implement a compareTo method like this:

@Override
public int compareTo(Object o) {
    if (!(o instanceof Patient))
        return -1;
    Patient temp = (Patient) o;
    int r = this.lastName.compareToIgnoreCase(temp.getLastName());
    if (r == 0)
        r = this.firstName.compareToIgnoreCase(temp.getFirstName());
    if (r == 0)
        r = this.SSN.compareToIgnoreCase(temp.getSSN());
    if (r == 0)
        r = this.dob.toString().compareToIgnoreCase(temp.getDOB().toString());
    if (r == 0)
        r = Integer.compare(this.getID(), temp.getID());
    return r;
}

I believe that would solve the problem you described.
I would advise you to read up (Javadoc or books) on TreeSet and HashSet and the importance of the equals, compareTo and hashCode methods.
If you want to put your objects in a Set or a Map, you need to know about these to implement that correctly.

Note
I based this compareTo method on your equals method.
You were comparing the date-of-birth by first calling toString. That’s not a very good way of doing that – you can use the equals method in java.util.Date directly. In a compareTo method the problem gets worse because dates do not sort correctly when you sort them alphabetically.
java.util.Date also implements Comparable so you can replace that comparison in the method with:

    if (r == 0)
        r = this.dob.compareTo(temp.getDOB());

In addition, if any of the fields could be null, you need to check for that as well.