Your outdated policy puts you in the position of deciding whose reasons for absence are legitimate. When an employee is out frequently enough, he or she cannot do his or her job adequately to meet the needs of the company. Update to a no-fault absenteeism policy which allows each employee to miss a specific amount of time for illness or personal reasons each fiscal year.
While you may still require the doctor's note when three or more consecutive days are missed for purported illness, you will also have a set limit in place. When an employee consumes his or her allotted time off at such a rate as to indicate that he or she likely will exceed the number of days allowed or when a certain percentage of allotted time off is taken, issue a warning.
Then, terminate those employees who surpass the number of days allowed. This will most likely rid you of the employees with chronic attendance issues sooner vs. later.
If an employee has a serious, legitimate illness, he or she will most likely provide much more detailed information without prompting, and you can determine how to deal with him or her most fairly utilizing family medical leave or personal unpaid leave of absence, or whatever applies to your size and type of business.