Is it possible to consider the occurrences of precipitation in two different places to be independent phenomena?
Depends quite a bit upon the geographic scale between the two places. But in general there will tend to be some interdependence, even for locations large distances apart, as the large-scale upper atmosphere is controlled by a wave pattern:
The most independent phenomena will tend to be mesoscale (local) phenomena like seabreeze thunderstorms and lake effect snow... though even they tend to have some fair reliance upon wind direction and pressure regime, which are involved with the larger pattern. The amount of dependence between two reasonably distant locations though will often be quite small.
On the other hand, it's quite common to have a few days where rain thousands of miles apart is not just slightly correlated, but directly the consequence of the same feature, such as this front from January 2013 that Wikipedia had saved:
So fundamentally there is some, at least small, interconnection between precipitation events at any two locations globally. But if you can choose locations with consideration of geographic (distance/surroundings) and meteorological factors (seasons), you can likely get results that are empirically basically independent, as the large-scale pattern connection between them is such a minor contributing factor compared to other influences. In other words, there is a dependent relationship part between the precipitation at any two locations and an independent part. Statistically the dependence is often very (very) small for many location choices, but is greater for nearer locations and in certain seasons/weather events.
If you are, for example, designing a school science fair experiment, you would likely struggle if you wanted to see which animal specie is best at predicting the weather (spoiler: none!) by looking at snowfall totals, because there's quite a lot of dependence between the weather at the sites listed since they are mostly in the same region and their winter/spring climate is heavily tied to the large-scale upper atmospheric pattern and storm system tracks, so that would likely swamp out any "contribution" from the animal (in addition, geographical differences, such as terrain and relationship to bodies of water, would be another huge uncontrolled independent factor). On the other hand, if you were wanting to compare the influence of industrial pollution on the number of rainfall days, and carefully chose, for example, distant tropical coastal sites (perhaps a site in SE Asia versus one in the Caribbean) with similar weather influences/patterns, and looked at large datasets... you might be able to fairly inspect the magnitude of such influence for the chosen regime type.
I often found meteorological phenomena to be challenging to design simple science fair projects around, as there are so many different interconnected variables, and it is hard to control most and leave one independent. That's why many meteorological studies are often focused more on multivariate analysis of large datasets and effects like urban heat island took quite a bit of careful analysis to recognize. It's not impossible, but to find two locations that are for all intents and purposes seeing independent weather events, while still not being controlled by other factors like their climate regimes/topography, can be quite challenging.
But while fundamentally no precipitation events are ever absolutely perfectly independent, with the right distance and location considerations their interdependence can be so negligible so as to consider the precipitation independent phenomena. I would consult a meteorologist to analyze how independent two such sites may be.