Nearly or Almost

Solution 1:

I almost hesitate to [tell you] - about 44,600 results in Google Books
I nearly hesitate to say - 5 results

...that there certainly are contexts where only one word is valid/likely to be used....

OP very nearly missed [the point] - about 16,100 results
He very almost missed it - no written instances

As to the reason for such marked differences in "acceptability", I think with the first pair it's because idiomatically we often use almost to indicate that the words following aren't quite to be taken literally. The second pair seems to suggest nearly can be qualified by other modifiers, whereas almost can't.