Time dilation due to gravity wells has been well tested by satellites in Earth's orbit. They aren't as deep in Earth's gravity well and experience less time dilation. It's very tiny, but measurable.

Even more, specific, the orbit of Mercury is affected by time dilation due to how close to the Sun it is in a very noticeable way. This was a real mystery - people speculated there had to be another planet pulling on it, named Vulcan - until Einstein proposed Relativity as the correct answer.