"To people in good health, young and strong, it does not much matter whether the rooms they inhabit be warm or cold, or damp or dry; but to others in delicate health, whose circulation is probably enfeebled by the accession of years, and mainly for that wide class of the sick who suffer from some bronchial or pulmonary affection, the [...] climate in-doors is of paramount importance. [...] Much less attention is given to the humidity of the air in-doors, for the very reason that we are hardly conscious of any change in this atmospheric condition." 
Curious piece of text, no? I just had to share it, not in small part because of the language used! It's the start of an an 1888 scientific paper about the so-called "bronchitis kettle", a device used in the 19th and 20th centuries to introduce more moisture into the air as a kind of over-the-counter treatment for chronic bronchitis. The earlier incarnations of the bronchitis kettle were designed for stoves, fireplaces, or lamps (which probably polluted the air quality all by themselves), while later versions relied on electricity. 
What Is Steam Inhalation Therapy?
Steam inhalation therapy is meant to help people with respiratory and sinus infections breathe more easily by loosening phlegm. People usually pour boiling hot water into a suitable container, add some essential oils, and hold their towel-covered head over the water and breath in the steam.
Is Steam Inhalation Therapy Effective? Can Steam Help With Bronchitis?
Numerous studies have been conducted into the efficacy of steam inhalation therapy. I found one that showed that steam inhalation therapy can indeed help — with the common cold . There are more, however, that have founded that it's impossible to conclude that the practice is objectively helpful for bronchiolitis , sinusitis , or the common cold . There's not, meanwhile, data on whether steam inhalation therapy can or can't help with chronic or acute bronchitis specifically.
What we do have is lots of people saying they feel better, subjectively, after trying steam inhalation therapy. That might be enough to induce you to have a go at it for yourself, but it's hardly solid science.
Why does the Heart, Lung, and Blood Institute say that "a humidifier or steam can help loosen mucus and relieve wheezing and limited air flow"  in people with bronchitis, then? "Steam inhalation therapy" may not have been proven effective, but introducing more moisture into your environment — the air you breathe in — certainly can:
- One study showed that having a humidifier in the home caused "all the people who experienced bronchitis [to experience] improvement in the disease" . It went on the emphasize the importance of maintaining this device properly, because, as another study found, improperly maintained humidifiers can easily disperse dust that will certainly not help your bronchitis .
- Going to the sauna — which is, let's face it, basically a giant steam inhalation therapy room — provides symptom relief to people suffering from asthma and chronic bronchitis , a study found.
- Harvard Medical School recommends hot, steamy showers to loosen phlegm if you have bronchitis .
Steam Therapy: To Try Or Not To Try?
Steam therapy alone won't "cure" your bronchitis. If it's acute, the viral or bacterial infection that caused it will need to run its course, or the allergen or irritant removed from your life. You will still need to do all the other things, like get plenty of rest, drink lots of water, eat a healthy diet, and take fever reducers. Depending on your exact situation, you may even be prescribed a bronchodilator, and antibiotics if you are dealing with a bacterial infection. If your bronchitis is chronic, well, it's there to stay, and prescription medications are likely to be a part of your life, while you can also reduce your symptoms with lifestyle changes, including taking a good look at volatile organic compounds that can aggravate bronchitis in your home.
Along with traditional steam inhalation therapy, trips to the sauna, using a (properly cared for) humidifier, and taking steamy showers might, however, be of some help to you. These moisture-inducing techniques are unlikely to cause you any harm, and so you don't have much to lose by trying. Never use any home remedy as a substitute for actual medical care, of course — and while you're good with plain water or chamomile, talk to your doctor if you're planning on adding (potentially irritating) essential oils to your steam therapy.