В конце концов я наткнулся на этот пост: http://w-shadow.com/blog/2007/08/12/how-to-force-file-download-with-php/ .

function output_file($file, $name, $mime_type='')
{
 /*
 This function takes a path to a file to output ($file), 
 the filename that the browser will see ($name) and 
 the MIME type of the file ($mime_type, optional).

 If you want to do something on download abort/finish,
 register_shutdown_function('function_name');
 */
 if(!is_readable($file)) die('File not found or inaccessible!');

 $size = filesize($file);
 $name = rawurldecode($name);

 /* Figure out the MIME type (if not specified) */
 $known_mime_types=array(
    "pdf" => "application/pdf",
    "txt" => "text/plain",
    "html" => "text/html",
    "htm" => "text/html",
    "exe" => "application/octet-stream",
    "zip" => "application/zip",
    "doc" => "application/msword",
    "xls" => "application/vnd.ms-excel",
    "ppt" => "application/vnd.ms-powerpoint",
    "gif" => "image/gif",
    "png" => "image/png",
    "jpeg"=> "image/jpg",
    "jpg" =>  "image/jpg",
    "php" => "text/plain"
 );

 if($mime_type==''){
     $file_extension = strtolower(substr(strrchr($file,"."),1));
     if(array_key_exists($file_extension, $known_mime_types)){
        $mime_type=$known_mime_types[$file_extension];
     } else {
        $mime_type="application/force-download";
     };
 };

 ob_end_clean(); //turn off output buffering to decrease cpu usage

 // required for IE, otherwise Content-Disposition may be ignored
 if(ini_get('zlib.output_compression'))
  ini_set('zlib.output_compression', 'Off');

 header('Content-Type: ' . $mime_type);
 header('Content-Disposition: attachment; filename="'.$name.'"');
 header("Content-Transfer-Encoding: binary");
 header('Accept-Ranges: bytes');

 /* The three lines below basically make the 
    download non-cacheable */
 header("Cache-control: private");
 header('Pragma: private');
 header("Expires: Mon, 26 Jul 1997 05:00:00 GMT");

 // multipart-download and download resuming support
 if(isset($_SERVER['HTTP_RANGE']))
 {
    list($a, $range) = explode("=",$_SERVER['HTTP_RANGE'],2);
    list($range) = explode(",",$range,2);
    list($range, $range_end) = explode("-", $range);
    $range=intval($range);
    if(!$range_end) {
        $range_end=$size-1;
    } else {
        $range_end=intval($range_end);
    }

    $new_length = $range_end-$range+1;
    header("HTTP/1.1 206 Partial Content");
    header("Content-Length: $new_length");
    header("Content-Range: bytes $range-$range_end/$size");
 } else {
    $new_length=$size;
    header("Content-Length: ".$size);
 }

 /* output the file itself */
 $chunksize = 1*(1024*1024); //you may want to change this
 $bytes_send = 0;
 if ($file = fopen($file, 'r'))
 {
    if(isset($_SERVER['HTTP_RANGE']))
    fseek($file, $range);

    while(!feof($file) && 
        (!connection_aborted()) && 
        ($bytes_send<$new_length)
          )
    {
        $buffer = fread($file, $chunksize);
        print($buffer); //echo($buffer); // is also possible
        flush();
        $bytes_send += strlen($buffer);
    }
 fclose($file);
 } else die('Error - can not open file.');

die();
}   

/*********************************************
            Example of use
**********************************************/

/* 
Make sure script execution doesn't time out.
Set maximum execution time in seconds (0 means no limit).
*/
set_time_limit(0);  
$file_path='that_one_file.txt';
output_file($file_path, 'some file.txt', 'text/plain');

Добавление всех рекомендованных здесь заголовков, а также использование:

 ob_end_clean(); //turn off output buffering to decrease cpu usage

перед любым выводом - помогло. Больше никаких ограничений не наблюдается. Файлы скачиваются полностью, даже огромные.