CloudFront with S3 website as origin is not serving gzipped files

My problem was that I uploaded the files specifically with utf-8 encoding. From the documentation:

CloudFront determines whether the file is compressible:

The file must be of a type that CloudFront compresses.

The file size must be between 1,000 and 10,000,000 bytes.

The response must include a Content-Length header so CloudFront can determine whether the size of the file is in the range that CloudFront compresses. If the Content-Length header is missing, CloudFront won't compress the file.

The response must not include a Content-Encoding header.

As Cloudfront now only accepts JSON, you have to paste this:

[
    {
        "AllowedHeaders": [
            "Authorization",
            "Content-Length"
        ],
        "AllowedMethods": [
            "GET"
        ],
        "AllowedOrigins": [
            "*"
        ],
        "ExposeHeaders": [],
        "MaxAgeSeconds": 3000
    }
]

I hit the same error today and solved it by adding a CORS rule to the S3 bucket. This rule ensures the Content-Length header is sent to Cloudfront so content can be gzipped:

S3 > Bucket > Permissions > CORS Configuration

<?xml version="1.0" encoding="UTF-8"?>
<CORSConfiguration xmlns="http://s3.amazonaws.com/doc/2006-03-01/">
    <CORSRule>
        <AllowedOrigin>*</AllowedOrigin>
        <AllowedMethod>GET</AllowedMethod>
        <MaxAgeSeconds>3000</MaxAgeSeconds>
        <AllowedHeader>Authorization</AllowedHeader>
        <AllowedHeader>Content-Length</AllowedHeader>
    </CORSRule>
</CORSConfiguration>

Credit goes to Robert Ellison: http://ithoughthecamewithyou.com/post/enable-gzip-compression-for-amazon-s3-hosted-website-in-cloudfront

As far I know, this seems to be an undocumented requirement.