| Maybe
Hopefully your neck simply woke up one humid morning and decided to bow backwards. You add a little neck relief by loosening the truss rod, and you're back to normal until the neck reacts to another drastic change in humidity, in which case you tweak the rod again.
Or there could be another, more sinister force at work. Most players wear down the the frets in the lower positions before they wear down the frets in the upper positions. As this process takes place, you begin to observe the symptoms that sent you to the keyboard, as opposed to the fret board.
Sometimes you can compensate for this with more neck relief, but this tends to be a compromise, since it gives you unnecessarily higher action in the higher positions. There comes a time in the lives of many instruments when a fret levelling/dressing is the best solution. This is a task best left to those who have done it succesfully a few times, especially if we're talking about your '62 Fender!
Compare the profile of your lower frets with the profile of frets further up the board, where you aren't having a problem. If the lower frets are starting to look flatter on top, or if the lower frets exhibit "divots" under the strings, then you probably need to have a pro give it a look, and discuss your options with him. | 
