How do I require one field or another or (one of two others) but not all of them?

The problem is the “not” semantics. “not required” does not mean “inclusion forbidden”. It just means that you don’t have to add it in order to validate that schema.

However, you can use “oneOf” to satisfy your specification in a simpler way. Remember that it means that “just one of these schemas can validate”. The following schema achieves the property switching you are attempting to solve:

{
    "$schema": "http://json-schema.org/draft-04/schema#",
    "type": "object",
    "required": [
        "unrelatedA"
    ],
    "properties": {
        "unrelatedA": {
            "type": "string"
        },
        "fileNames": {
            "type": "array"
        },
        "copyAll": {
            "type": "boolean"
        },
        "matchesFiles": {
            "type": "array"
        },
        "doesntMatchFiles": {
            "type": "array"
        }
    },
    "oneOf": [
        {
            "required": [
                "copyAll"
            ]
        },
        {
            "required": [
                "fileNames"
            ]
        },
        {
            "anyOf": [
                {
                    "required": [
                        "matchesFiles"
                    ]
                },
                {
                    "required": [
                        "doesntMatchFiles"
                    ]
                }
            ]
        }
    ]
}