There are multiple verses warning the saints to be careful not to fall away, which implies that it's possible for any of us--even the elect--to fall away.
Actually, the verse goes like this:

"For false Christs will arise, and false prophets, and will show great signs and wonders; so that, if it were possible, they would deceive the very elect."

Some take that to mean it's not possible.

There are verses that seem to support perseverance of the saints, like John 10.

27 My sheep hear my voice, and I know them, and they follow me. 28 I give them eternal life, and they will never perish, and no one will snatch them out of my hand. 29 My Father, who has given them to me, is greater than all, and no one is able to snatch them out of the Father's hand.