In England at least, yes! We do have archaeological evidence of cisterns being used for collecting drinking water in countryside villages. However, that evidence shows that since access to water was a prime concern for where villages cropped up along the countryside, wells were the most common source of water, when a village wasn't near a river for streams. We know this both simply because many of the villages that still exist have historic wells, along with the fact that we have court records of accidental deaths that occurred while drawing well water (it was uncommon, but it happened). Incidentally, women seem to have done most of the water-drawing.
Rainwater is always good, but while living in Great Britain might make you think otherwise, the rain isn't always reliable. It's much simpler to get water from constant sources.
Also Venice, which historically has had no access to spring water or rivers, did rely on large cisterns almost entirely for fresh water. They had lovingly-designed Byzantine-inspired cisterns called vere da pozzo as far back as the 11th century. They used them like well-heads, letting rainwater filter underground in the campi (fields) into those cisterns, filtering through gravel and sand, before they could reach the salt water around them. One of these style cisterns was found in Toledo, Spain, too, so cisterns do indeed seem to have made their way around medieval Europe as important water-sources.
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