S3 PutObject rate limit reached

I have a S3 bucket in which keys are organized as

<user_id>/<dataset_id>/<actual data blocks>

<user_id> may take ~3250 different values. <dataset_id> are always unique.

Every second, my service write to each prefix with a data block of ~30Kb. Then I got the following error:

(SlowDown) when calling the PutObject operation (reached max retries: 4): Please reduce your request rate.

The documentation clearly stated that S3 supports "3,500 PUT/COPY/POST/DELETE or 5,500 GET/HEAD requests per second per prefix", but looks like it's not the case. S3 cannot even handle 1 request per second with 3250 concurrent writers.

Is there any problem in the way I organize my bucket keys or it's the S3 capacity/documentation issue? Thanks!