Arduino sketches not working - arduino

Question

I've got an Arduino r3 and a Arduino GPS-shield 2.
The example-sketches and the code on arduino.cc doesn't work.
None of the tips I've read about, both here on SO and everywhere else, doesn't work for me.
I've tried the AT command tester to see if it can handle AT commands, and just get "Invalid or no response from the device". (I've set the right port and baud rate (tested 19200 and 9600))
I've even bought new parts to eliminate faulty boards...
For example:
// libraries
#include <GSM.h>
// modem verification object
GSMModem modem;
// IMEI variable
String IMEI = "";
void setup()
{
// initialize serial communications and wait for port to open:
Serial.begin(9600);
while (!Serial) {
; // wait for serial port to connect. Needed for Leonardo only
}
// start modem test (reset and check response)
Serial.print("Starting modem test...");
if (modem.begin())
Serial.println("modem.begin() succeeded");
else
Serial.println("ERROR, no modem answer.");
}
void loop()
{
// get modem IMEI
Serial.print("Checking IMEI...");
IMEI = modem.getIMEI();
// check IMEI response
if (IMEI != NULL)
{
// show IMEI in serial monitor
Serial.println("Modem's IMEI: " + IMEI);
// reset modem to check booting:
Serial.print("Resetting modem...");
modem.begin();
// get and check IMEI one more time
if (modem.getIMEI() != NULL)
{
Serial.println("Modem is functoning properly");
}
else
{
Serial.println("Error: getIMEI() failed after modem.begin()");
}
}
else
{
Serial.println("Error: Could not get IMEI");
}
// do nothing:
while (true);
}
Serial monitor says:
Starting modemtest... ERROR, no modem answer.
Checking IMEI...Modem's IMEI: 0
Resetting modem...Modem is functioning properly.
I've followed every lead I have found (for almost a week) on a whole bunch of forums, but it's still not working. What have I missed?