My Ancestry composition at 23andMe states:
"You most likely had a French ancestor, second, third, fourth or fifth great grandparent who was 100% French born between 1760 and 1850."
How accurate is that date likely to be?
It's statistics. "Accurate" simply doesn't apply: a prediction may be 99% likely, but that will not mean a thing to that 1 in 100 where it fails -- and you have no way of knowing which group you fall into.
More specifically, the ancestry timeline on 23andMe makes certain underlying assumptions that may or may not be true.
One assumption is that their admixture calculators have accurately labeled your ancestral origins. I would argue that with the current state of the DNA databases, this is simply not true. They cannot tell near-neighbor populations apart, and there are DNA patterns that they've misidentified as belonging to certain populations, when in fact those same patterns occur in other populations that simply aren't represented in their reference samples (yet).
Another assumption is that the nearby DNA variants that the report is based on came from a single ancestor. 23andMe's "white paper" (https://permalinks.23andme.com/pdf/23-14_admixture_date_estimator.pdf) on their ancestry timeline feature states this as: "we assume a model where exactly one ancestor contributed an ancestry", and they admit that this is simplistic.
There is a third assumption at play, but it's an "aftermarket" one, made by customers, based on the associations between 23andMe's chosen labels for admixture groupings and modern geopolitical boundaries. For example, if you saw "You most likely had an ancestor who was 100% French & German", you'd probably conclude that you're supposed to be looking for an ancestor in France or Germany, right? But in fact, due to centuries of migration, that "French & German" ancestor (if he/she existed) could've lived anywhere from Ukraine to Utah.
Also, regarding the dates: 23andMe uses 30 years as an average generation, using ever-broadening ranges of dates as they go back in time in order to account for the differences possible. (Their somewhat-awkward phrasing: "When someone’s great-grandparents lived is quite different whether that someone is 9 or 90.") But their broadening ranges are not quite broad enough to account for a grandparent born a full century earlier than his grandchild (as is the case for my spouse).