Is a dystopian future with a thinner atmosphere and heavy snow feasible? Also would that mean that snow clouds formed at lower altitudes?
I would start with the magnetosphere. For example Mars has a very weak magnetosphere therefore much of the atmosphere has been lost to solar wind. If you could somehow dramatically alter the composition or properties of the Earth's core to be less magnetic then you could make the Earth more susceptible to solar winds which could strip away parts of the atmosphere.
This however would cause other problems as this would also make the Earth more susceptible to harmful solar and cosmic radiations, which is going to be a problem for most forms of life.
This question is very complex because there are different parts of the atmosphere which serve different purposes and would have different consequences if altered. "Thinning out " could also mean to extend our current atmosphere further out into space therefore making it less dense. This is much more difficult to conceive then stripping away mass from the atmosphere. Hopefully this can give you enough of a starting point to further your own research. conversely your could also have exceptionally strong solar winds occurring for a similar effect--without having to deal with the Earth's own magnetic properties.
EDIT: My answer is somewhat off the cuff due to writing from my phone but here is an article which details the concepts I am referring to: http://www.nasa.gov/press-release/nasa-mission-reveals-speed-of-solar-wind-stripping-martian-atmosphere
I would point out that the atmosphere doesn't need to be lost to thin out. The atmospheric composition has varied widely over time based almost entirely on gases becoming locked up in soil and crustal rock. The mass of the atmosphere is roughly 10,000 times smaller than the mass of the crust, meaning tiny changes in the way that gases become sequestered can make huge differences in the atmosphere.
You might invent a new form of oceanic plant life which is a prolific nitrogen fixer, and composed of a substance which is very difficult for oceanic microorganisms to break down.
Just this type of scenario happened during the carboniferous period. During this time plants began producing lignin and suberin in large quantities for the first time. Microorganisms were unable to break down these substances before they became buried. This buried organic material contained huge quantities of carbon (eventually becoming much of the coal available to us today), meaning that the levels of carbon dioxide in the atmosphere plummeted, oxygen levels soared to 45%, and the climate cooled 8 degrees C.
I suggest nitrogen in particular because it is 78% of the current atmosphere.
I think the phrase you will want to google is "nitrogen sequestration". Here is one example of an article discussing how nitrogen is sequestered currently. https://www.sciencedaily.com/releases/2015/12/151201115037.htm
"The evil leaders of Planet Spaceball, having foolishly squandered their precious atmosphere, have devised a secret plan to take every breath of air away from their peace-loving neighbor, Planet Druidia."
In all seriousness, there a only a few ways I can think of as to why an object's atmosphere would be lost.
- A large amount of gravitational force is applied to it by an enormous object nearby. This frequently happens in binary star systems or when a star orbits a pulsar, as shown in the image.
(Original Source: http://www.castfvg.it/stelle/pulsar/pulsar_001.jpg)
This is highly unlikely to happen, as it would take an object like a super-earth or a gas giant moving close to the planet to have enough gravity for this to occur. If said planet was on a highly elliptical orbit, it would extremely infrequently go close enough to any other planet to effect them gravitationally. when it comes to this way, it would be a really rapid event, causing incredible winds, massive earthquakes, or even total destruction through tidal forces if the giant was big/close enough. Most spectacular, but I would not recommend due to the probable extinction of all life on the planet.
- A large, un-oxidised object (or series of objects) lands on the planet, and much of the oxygen in the atmosphere is adsorbed by oxidation. This will require a metallic asteroid(s), and an enormous amount of material. This has the additional effect of reducing the oxygen content in the atmosphere.
If your planet already had a higher percentage of oxygen, this is not as much of a problem.
Either way, your planet will suffer extremely reduced greenhouse effect, making nights especially cold.
In relation to snow and precipitation, water will evaporate more readily under lowered pressure, and the clouds will form at a lower altitude due to the decrease in temperature. This should lead to pretty impressive snowfalls, unless it gets cold enough that the oceans freeze over.