The book of Daniel was almost certainly written in the 2nd century BCE, probably based on some historical accounts mixed with Jewish folklore. With that in mind, and given that there isn't any evidence that Nebuchadnezzar went insane for seven 'times', is there any other period from which the reference to "seven times" at Daniel 4:13-17 may be derived?
Daniel 4:13-17:
“‘I continued beholding in the visions of my head upon my bed, and, look! a watcher, even a holy one, coming down from the heavens themselves. 14 He was calling out loudly, and this is what he was saying: “CHOP the tree down, and cut off its boughs. SHAKE off its foliage, and scatter its fruitage. Let the beast flee from under it, and the birds from its boughs. 15 However, LEAVE its rootstock itself in the earth, even with a banding of iron and of copper, among the grass of the field; and with the dew of the heavens let it be wet, and with the beast let its portion be among the vegetation of the earth. 16 Let its heart be changed from that of mankind, and let the heart of a beast be given to it, and let seven times pass over it. 17 By the decree of watchers the thing is, and [by] the saying of holy ones the request is, to the intent that people living may know that the Most High is Ruler in the kingdom of mankind and that to the one whom he wants to, he gives it and he sets up over it even the lowliest one of mankind.”
The JWs' interpretation of "seven times" as '2520 years' is based on numerological gymnastics without any clear precedent, and can therefore be discounted. So it would seem that "seven times" refers to either some other period or nothing at all.
If it does refer to an actual period, then it would likely be based on terms with which the Jews were familiar. The most likely candidate is found at Leviticus chapters 25 and 26. (Leviticus is generally considered to have been completed in the Persian period after 538BCE.)
Leviticus 25:8:
"“‘And you must count for yourself seven sabbaths of years, seven times seven years, and the days of the seven sabbaths of years must amount to forty-nine years for you.
Leviticus 26:27-35:
“‘If, however, with this YOU will not listen to me and YOU just must walk in opposition to me, 28 I shall then have to walk in heated opposition to YOU, and I, yes, I, shall have to chastise YOU seven times for YOUR sins.
Leviticus 26:32-35:
32 And I, for my part, will lay the land desolate, and YOUR enemies who are dwelling in it will simply stare in amazement over it. 33 And YOU I shall scatter among the nations, and I will unsheathe a sword after YOU; and YOUR land must become a desolation, and YOUR cities will become a desolate ruin.
34 “‘At that time the land will pay off its sabbaths all the days of its lying desolated, while YOU are in the land of YOUR enemies. At that time the land will keep sabbath, as it must repay its sabbaths. 35 All the days of its lying desolated it will keep sabbath, for the reason that it did not keep sabbath during YOUR sabbaths when YOU were dwelling upon it.
For how long was Jerusalem desolate (or For how long was the 'tree' in the dream banded)? Jerusalem was destroyed in 587BCE, and the Jews (but not all of them) returned to Jerusalem in 538BCE. Seven 'times'. Seven Sabbaths. Forty-nine years.