Paul's warning about "the last days" would be strange if he were merely describing the normal brokenness that has been common to man since the dawn of time. (Such brokenness would not be news to anyone, and would not be specific to "the last days.") He seems to be anticipating something unusual. ...And yet, the characteristics he is describing have been common to man since the dawn of time -- at least to some extent. Perhaps the best solution is to see Paul's statement as an anticipation of the present condition getting "much worse" prior to the end.
Thus, Paul could expect Timothy to be familiar with "such people" in Timothy's present situation, but that would not imply that Timothy's present situation was equivalent to the escalated state of things that Paul anticipated just prior to the end.
Note: "The last days" is a period of time referenced throughout Scripture, which began with the coming of Christ, and will continue until Christ rules on earth and subdues His enemies. The focus of the prophecies are on the Christ, and so from a Christian perspective, many focus on events that occurred 2,000 years ago, and many focus on events yet to come, but technically the age we live in also counts as "the last days."
So Timothy did technically live in "the last days" (see below), but that does not imply that all prophecy had therefore been fulfilled, as if "the end times" had already come and gone.
At Pentecost, Peter explains what they are seeing by quoting a prophecy:
'And it shall be in the last days,’ God says, ‘That I will pour forth of My Spirit on all mankind; And your sons and your daughters shall prophesy, And your young men shall see visions, And your old men shall dream dreams' -Acts 2:17
Peter's point is that "the last days" is now, and that what was prophecied is what they are seeing. The author of Hebrews makes similar use of the term:
God, after He spoke long ago to the fathers in the prophets in many portions and in many ways, in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world. -Hebrews 1:1-2
He is referring, of course, to Jesus coming in the flesh. His point is that "long ago" (prior to Christ) God spoke through the prophets, but "in these last days" He spoke to us by sending His Son. This implies that "the last days" began with the coming of Christ.