Maltese can be prone to two respiratory disorders; Reverse sneezing and Collapsing Trachea. One is harmless while the other can become a serious problem with age. The symptoms for both can be similar even though they are totally different conditions. It's important to discern which is which in case treatment is required.
Collapsing Trachea on the other hand is a progressive disease that gets worse with age. In a normal dog the windpipe is a tube made up of cartilage rings allowing air to pass in and out. In some dogs, the rings for some reason lose their 'stiffness' and will collapse, partially obstructing the airway. Symptoms will include a dry harsh cough that is often described as a 'goose honk'. Often any kind of pressure on the trachea will elicit this cough. The dog will experience exercise intolerance. Excitement or stress will also trigger a coughing spell. Unlike reverse sneezing, which begins at a young age, clinical signs for collapsing trachea don't usually start exhibiting themselves until middle age (but that is not a hard fast rule, it can affect dogs at any age). Diagnosis is made by X-rays or endoscopy. Most vets will try medical management to try to control the cough, inflammation and open up the airway. There is no cure, but if the symptoms aren't too severe, this can be quite successful giving the dog good quality of life. Secondary problems can result from a collapsing trachea, so it is important that the dog have regular checkups by the vet. If the symptoms become severe, the option for surgery might need to be explored. There seems to be little information as to whether or not collapsing tracheas are inherited. But given that it occurs in most toy breeds (probably due to downsizing), it can be assumed that there is a genetic tendency to developing it.
Certainly affected dogs should never be bred with this disease.