It appears to me that in the survey by Gottesman (around Thm 2) as well as the book by Nielsen and Chuang (Thm 10.2) it is suggested that if a QEC code corrects errors $A$ and $B$ then it also corrects any linear combination of errors (in particular by Gottesman); the sources can be found here:
Gottesman: https://arxiv.org/abs/0904.2557 Nielsen, Chuang: http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf
A simple QEC code like Shor's 9-qubit code can correct arbitrary single-qubit errors bc it can correct the Pauli errors if they occur on the same qubit, but clearly it cannot correct more than one error if they occur in the wrong places (e.g. two bitflip errors in the same block). But such an error would be a linear combination of a bitflip error X_1 hitting the first and a bitflip error X_2 hitting the second qubit in the code. What am I missing here?