No. A late date for the earliest manuscript of any ancient book could be evidence that the book is a forgery or unreliable;1 but it does not necessarily follow that a late date for an earliest manuscript means the book is a forgery or is unreliable. There is other evidence to take into account.
Some of the evidence that needs to be taken into account, for example, are the quotes and allusions that early Christians take from books of the Bible. The earliest manuscript we have for 1 and 2 Timothy are from c. 350 AD. However, early Christians quote from the letters demonstrating that they have been part of Christian history from before c. 350 AD. Examples can be seen in Polycarp’s Letter to the Philippians (Polycarp lived c. 69 – 155):
“But the love of money is the root of all evils.” Knowing, therefore, that “as we brought nothing into the world, so we can carry nothing out,” let us arm ourselves with the armour of righteousness; [ch. 4]
Here Polycarp quotes from 1 Tim. 6:10 and v7.
He [the Lord] has promised to us that He will raise us again from the dead, and that if we live worthily of Him, “we shall also reign together with Him,” provided only we believe. [ch. 5]
Here Polycarp quotes from from 2 Tim. 2:12.
Other evidence is in the form of canon lists and early translations in other languages. For example, continuing to use Paul’s letters to Timothy as our case study, the Muratorian Canon (c. 170-200 AD) mentions ‘two [letters] to Timothy’.
Both Polycarp and the Muratorian Canon pre-date the 350 AD manuscript of 1 and 2 Timothy, demonstrating that a late date for an earliest manuscript of a book doesn’t mean the book is either unreliable or a forgery.
1. It’s interesting to note that rarely do people lay this charge (that a late date for an earliest manuscript equals unreliability or forgery) at the feet of non-biblical books like Homer’s Iliad (even though the date between the original composition of the Iliad and the earliest manuscript of the book is large [~500 years] — larger than the date between the original composition of the NT books and their earliest manuscripts). Perhaps there’s something about the claims of the Bible that cause people to want to treat it differently (and, perhaps it could be said, unfairly).