When should you be worried when your pet is coughing?
First you have to distinguish coughing from other sounds you pet could make. Sneezing and snoring are easy to distinguish. Reverse sneezing is a sound a dog makes when the soft plate is in the wrong position, it is an inspirational sound.
Coughing can be one big cough, for instance if the dog drank water too quickly. Or a couple of harsh coughs if something got stuck in the throat. This is functional and will seize as soon as the throat is cleared.

Coughing can be dry, persistent and not productive. These are irritations of the throat and trachea. In dogs commonly caused by kennel cough. Kennel cough is a complex syndrome caused by para-influenza viruses and the bordetella bacteria. It is very contagious; it just needs one dog coughing in the direction of another dog to transmit the disease. Hence the name kennel cough. It is an infection of the larynx and trachea and will need treatment. Cats could be coughing due to hair in the throat, although they rather tend to vomit. Cats could suffer from feline asthma, which could be due to allergy, which makes them cough dry and harsh.
Coughing can be productive, which means mucus needs to come up. This is due to bronchitis. Beware that dogs and cats will cough up the mucus but then swallow it in, so you do not see it. Bronchitis needs treatment for the infection and products that make it easier to cough mucus up. Cat flu can progress to bronchitis.
Coughing can be present only after exercise or only at night. This is highly suggestive of heart disease. This cough is not productive and often these dogs pant a lot. They have a low exercise tolerance. Diagnostics like blood tests, x rays, ultrasound, ECG need to be done to determine the type of heart disease and medications will be given.
If you notice your pet, particularly your cat, having breathing difficulties but is not coughing at all, please visit your veterinarian as soon as possible. Either a severe pneumonia, a collapsed lung or another serious lung disease could be the cause of this.
Old geriatric dogs and cats could be coughing because of lung disease like fibrosis or tumors.
This could be a primary tumor in the lung or metastatic lung disease.
Sometimes pets (cats!) do not cough when at the vets, or on the examining table, therefore it is always a good idea to do a video recording at home and bring that with you. The body position during the coughing can be important as it shows the level of respiratory distress.