ssize_t, size_t readn, writen function from unix network programming

Question

there is this function from the book "Unix network programming where there is a function "writen" declared as following:

ssize_t writen(int fd, const void *vptr, size_t n){
    ssize_t nleft, nwritten;
    const char *ptr;

    ptr = vptr;               
    nleft = n;
    while (nleft > 0) {
        if ((nwritten = write(fd, ptr, nleft)) <= 0)
            return nwritten;  // error

        nleft -= nwritten;
        ptr += nwritten;
    }
    return n;
}

From what I checked in linux manual page:

size_t
Used for a count of bytes. It is the result of the sizeof() operator. It is an unsigned integer type capable of storing values in the range [0, SIZE_MAX].

ssize_t
Used for a count of bytes or an error indication. It is a signed integer type capable of storing values at least in the range [-1, SSIZE_MAX].

but {SIZE_MAX} = 65535 and SSIZE_MAX >= 32 767

Does this mean that whe using writen with n > 32767 I could get an overflow and the function would return fewer bytes then it actually had writen to the descriptor?