By a yet undiscovered set of circumstances some records are created without addresses. Because the CRM has to create two default related address records, editing the master records will produce errors that invoke internal business rules of the CRM.
To solve it, navigate to the More Addresses and add two related records. Don't worry when they do not show up in the grid, they are actually being created as the default address set. If you want to see it in action, save the first one and reload the page. The info will be shown in all of the Address 1 fields; same thing for the second one and Address 2 fields.
After you have created both, editing of the problematic record can be resumed.