API fails to autorename the files with large file size

Akshata, no, the expected behavior is that files of all sizes work the same way. That said, I'm unable to reproduce the issue with a 13-byte file. When I try it, the file remains unchanged and not renamed:

$ echo Hello, World | http post https://content.dropboxapi.com/2/files/upload Dropbox-Api-Arg:'{"path": "/test", "mode": "add", "autorename": true}' Content-Type:application/octet-stream Authorization:'Bearer <REDACTED>'

HTTP/1.1 200 OK
...
Date: Fri, 27 May 2016 04:34:27 GMT

{
    ...
    "name": "test",
    "path_display": "/test",
    "path_lower": "/test",
    "server_modified": "2016-05-27T04:34:27Z",
    "size": 13
}

$ echo Hello, World | http post https://content.dropboxapi.com/2/files/upload Dropbox-Api-Arg:'{"path": "/test", "mode": "add", "autorename": true}' Content-Type:application/octet-stream Authorization:'Bearer <REDACTED>'

HTTP/1.1 200 OK
...
Date: Fri, 27 May 2016 04:34:33 GMT

{
    ...
    "name": "test",
    "path_display": "/test",
    "path_lower": "/test",
    "server_modified": "2016-05-27T04:34:27Z",
    "size": 13
}

Please double check that you're actually sending the same data both times.