True. Tom while he was evasive, didn't discourage Michael in Vegas. Also nobody knew then Fredo's deeper involvement.
It was when Michael learned that Pentangeli was alive

Tom after failing to be on top of the Pentangeli survival and arrest, says he doesn't know anything -- and I believe him like Michael won't get anything from him. Then we find Fredo knew plenty though nothing of value to Michael

Michael no doubt thinking he was invulnerable, sitting high up in the Sierra mountains, drinking Champaign, Uh, Champaign Cocktails --

Don't know taking the Fifth was the safe route for the “legitimate” Don Michael Corleone
and if Tom advised him that then he didn't know Michael, didn't know that that's an impossibility -- that could never happen as to why he didn't, couldn't, in fact never could take the Fifth Amendment. Nothing to hide

Michael looms large in everything and we have strayed from your original question!
I reckon, there were no Good or bad feelings, competition, suspicion, etc. with each other but
1. They all were fiercely protective and guarding their own territory, to hang his hat
2. I can't see Neri taking orders from Tom though
3. As we saw Rocco wasn't happy! taking orders from Tom