I"m sorry to hear of this with Max. My initial thought was feline psychogenic alopecia which certainly can exist but her being "sick quite a lot..." suggests an underlying food intolerance as well.
Feline psychogenic alopecia is overdiagnosed but does exist. Excessive and out-of-context grooming is thought to be an obsessive-compulsive behavior that's triggered by environmental stresses and anxiety. The condition is uncommon in cats, with purebred cats that have high-strung nervous temperaments being possibly predisposed. Flea hypersensitivity, food allergy, atopy, and other ectoparasites are more common causes of feline alopecia.
The alopecia of feline psychogenic alopecia may occur anywhere on the body where the cat can lick but it most commonly involves the medial forelegs, inner thighs, perineum, and ventral abdomen. Hair loss is often bilaterally symmetrical but remaining hairs don't epilate easily.
The underlying cause of the psychological stress (e.g., separation from owner, moved to a new house, animal companion died, new pet in household, formerly outdoor cat denied access to outdoors) must be identified and appropriate environmental modifications made, if possible. A good flea control program should be instituted to prevent fleas from aggravating the symptoms. Use of a mechanical barrier (e.g. Elizabethan collar, T-shirt) for 1-2 months to prevent grooming may help break the habit. Behavior-modifying drugs may help stop the abnormal grooming behavior. In some cases, treatment may be discontinued after 30-60 days of therapy; in others, lifelong therapy is required for control. Drugs that may be effective include the following: amitriptyline, clomipramine, buspirone, phenobarbital, diazepam, and naloxone.
Food intolerance/allergy is addressed with prescription hypoallergenic diets. These special foods contain just one novel (rabbit, duck, e.g.) animal protein or proteins that have been chemically altered (hydrolyzed) to the point that Max's immune system doesn't "see" anything to be allergic to. The over the counter hypoallergenic foods too often contain proteins not listed on the label - soy is a common one - and these proteins would confound our evaluation of the efficacy of the hypoallergenic diet. The prescription foods are available from her vet. There are many novel protein foods and a prototypical hydrolyzed protein food is Hill’s Prescription Diet z/d ultra (a hydrolyzed protein diet is my preference because it avoids the possibility of my patient being intolerant to even a novel protein). A positive response is usually seen within a few weeks if we’ve eliminated the offending food allergen. Food intolerance can arise at any age and even after our patient has been eating the same food for quite some time.
Please respond with further questions or concerns if you wish. I would consider a hypoallergenic food trial before medicating her. What do you think?