Java Apache Commons Net SMTP Wrapper Attachment
SRP - single responsibility principle  

The code is violating the SRP, as it does to many things.  

It is  


adding/composing headers for the attachments  
reading files from disc  


I would extract the composition of the headers to a separate method. 

private String getHeaders() {
    StringBuilder sb = new StringBuilder();

    sb.append("\r\n").append("--").append(bound).append("\r\n");
    sb.append("Content-Type: application/octet-stream\r\n");
    sb.append("Content-Transfer-Encoding: base64\r\n");
    sb.append("Content-Disposition: attachment; filename=\"");

    return sb.toString();
}  


and would assign the result before the for loop to a String variable.  

Next we extract the file reading to a separate method. I don't know if Android supports Try with resources so I don't use it here. We also rename the int nRead to bytesRead.  

private byte[] readFile(File file) throws FileNotFoundException, IOException {
    FileInputStream fileInputStream = new FileInputStream(file);
    ByteArrayOutputStream buffer = new ByteArrayOutputStream();
    int bytesRead;
    byte[] data = new byte[1024];

    while ((bytesRead= fileInputStream.read(data, 0, data.length)) != -1) {
        buffer.write(data, 0, bytesRead);
    }

    buffer.flush();
    fileInputStream.close();

    return buffer.toByteArray();
}  


next, we add a method which takes a file and returns the content as base64 encoded string  

private String getContentAsBase64(File file) throws FileNotFoundException, IOException {

    byte[] fileContent = readFile(file);

    return Base64.encodeBase64String(fileContent);
}  


After using the fluent append() method instead of the string adding using +, this leads to the former code beeing refactored to  

Writer w = client.sendMessageData();
String attachmentHeader = getAttachmentHeader();
for (File file : myAttachments) {
    w.append(attachmentHeader);
    w.append(Tools.getNameForAttach(file.getName()));
    w.append("\"\r\n\r\n");
    w.append(getContentAsBase64(file));
    w.append("\r\n");
}
w.close();