I can see that even in a simple scenario without multiple IRAs and multiple custodians: my spouse's traditional IRA (he only has one). He's rolled over 401K plans into it as he's had different employers and he could have had a basis in these contributions. Also, he's had Roth conversions which used up some of the basis. It would be a bit much for Fidelity or another custodian to track these. When the IRA was first started though, he did get a statement from Fidelity showing the initial basis which they got from his employer plan.
ETA: I can see in the situation with multiple IRAs (aggregate), the custodian only knows what you took out of their particular holding. They don't know the what you possible took out of other IRAs or the basis in those as well.
