Have a question about an issue that popped up yesterday. I don’t have the greatest theory, but I’m also not totally unaware.
We were playing in, I believe Am, and there was a Dm chord. I threw in a third (F) and the lead stopped me and said that that is wrong and if I ever play a third it should be from the key of the song, not the chord we are on.
I was a bit confused because all of bass theory I’ve learned, especially fifths, have been played off of the current chord.
I tend to fall in with the “there are no wrong notes” camp. Granted, I may just be using that as an excuse for my crappy playing but I digress. Here is Lord Wooten demonstrating this:
Two of my favorite quotes from the video:
“They don’t hire me to play the right key. They hire me to make you groove.”
“If I groove hard enough, I can play any note and you will like it. I can play in the wrong key so well that I can make you sound wrong.”
I’m sure there will be some other more complete answer, but I’ll get my two cents in. As usual, it depends. I would say that a third of the chords is good, for walking bass lines and general riff in most styles I can think of.
In some cases - not this one if you remember correctly - the third of a chord doesn’t belong to the key you are playing in, one of the most common would be when playing E (instead of Em, the diatonic V) in the Am key. However, I would go with the major third (G#) in such a case.
I am sure there are exceptions to the rule and to every exception, so without knowing the full context, it is difficult to say for sure what they meant, and if somebody is “more right”.
Still, to me, it looks like you weren’t doing anything particularly wrong
F is in Am, and not only that, is one of the most important notes, one that differentiates Am as a separate key from its relative major - it’s the minor sixth. It’s a very important note in the key and I cannot think of a reason to avoid it. Dm is correct in a progression from Am as well.
You’re at least off by a third, if you want to play it safe.
Fascinating topic, I have no answer - other than that it has happened to me too. Maybe the Dm and your F isn’t the thing. As in, the other instruments were doing something that clashed with the F, or you were playing the F when the melody had moved on. I guess the lead wanted something else.
The one thing I do know is that explanations are not the truth - the lead might make sense in their own mind, but not to you.
The only questions are WHEN the F (third of Dm) was played, and for what duration.
The sixth of a scale can be very complimentary when played as part of a line, but where it falls in the line, and what follows it, makes or breaks the worthiness of its usage.
Is it possible that the guitarist (I assume) made a mistake and played a D (or if not a mistake, a deliberate deviation from the standard progression) rather than a Dm? In which case he would have shifted out of key slighty, and the flat 3rd would have clashed more? Just an off the top of my head thing that came to mind when I first read the op.
You’re at least off by a third, if you want to play it safe. 
