In a bash script, using the conditional "or" in an "if" statement

If you want to say OR use double pipe (||).

if [ "$fname" = "a.txt" ] || [ "$fname" = "c.txt" ]

(The original OP code using | was simply piping the output of the left side to the right side, in the same way any ordinary pipe works.)

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After many years of comments and misunderstanding, allow me to clarify.

To do OR you use ||.

Whether you use [ or [[ or test or (( all depends on what you need on a case by case basis. It's wrong to say that one of those is preferred in all cases. Sometimes [ is right and [[ is wrong. But that's not what the question was. OP asked why | didn't work. The answer is because it should be || instead.

The accepted answer is good but since you're using bash I'll add the bash solution:

if [[ "$fname" == "a.txt" || "$fname" == "c.txt" ]]; then

This is documented in the bash reference manual 3.2.4.2 Conditional Constructs

expression1 && expression2

True if both expression1 and expression2 are true.

expression1 || expression2

True if either expression1 or expression2 is true.

The && and || operators do not evaluate expression2 if the value of expression1 is sufficient to determine the return value of the entire conditional expression.