I'm starting to think this must be a hardware problem.
I'm now using the sketch below to test this display. The LCD is from DX.com, part number GY-LCD-V1. This thread gives the code for setting the screen up: http://arduino.cc/forum/index.php/topic,106655.0.html.
This sketch is based on the sketch in that thread. It basically prints the number of seconds since startup to the serial line and the LCD display. The serial line keeps counting, but the screen gets about as far as 4 or 6 and then just stops. Pressing the reset button on the board restarts the sketch and the serial line begins counting from 1, but the screen won't re-initialise - I have to remove the USB cable to do that.
//DFRobot.com
//Compatible with the Arduino IDE 1.0
//Library version:1.1
#include <Wire.h>
#include <LCD.h>
#include <LiquidCrystal_I2C.h> // F Malpartida's NewLiquidCrystal library
#include <Time.h>
#define I2C_ADDR 0x20 // Define I2C Address where the PCF8574A is
#define BACKLIGHT_PIN 7
#define En_pin 4
#define Rw_pin 5
#define Rs_pin 6
#define D4_pin 0
#define D5_pin 1
#define D6_pin 2
#define D7_pin 3
#define LED_OFF 0
#define LED_ON 1
#define printByte(args) write(args);
uint8_t bell[8] = {0x4,0xe,0xe,0xe,0x1f,0x0,0x4};
uint8_t note[8] = {0x2,0x3,0x2,0xe,0x1e,0xc,0x0};
uint8_t clock[8] = {0x0,0xe,0x15,0x17,0x11,0xe,0x0};
uint8_t heart[8] = {0x0,0xa,0x1f,0x1f,0xe,0x4,0x0};
uint8_t duck[8] = {0x0,0xc,0x1d,0xf,0xf,0x6,0x0};
uint8_t check[8] = {0x0,0x1,0x3,0x16,0x1c,0x8,0x0};
uint8_t cross[8] = {0x0,0x1b,0xe,0x4,0xe,0x1b,0x0};
uint8_t retarrow[8] = { 0x1,0x1,0x5,0x9,0x1f,0x8,0x4};
LiquidCrystal_I2C lcd(I2C_ADDR,En_pin,Rw_pin,Rs_pin,D4_pin,D5_pin,D6_pin,D7_pin);
void setup()
{
Serial.begin(9600);
lcd.begin (20,4); // initialize the lcd
// Switch on the backlight
lcd.setBacklightPin(BACKLIGHT_PIN,NEGATIVE);
lcd.setBacklight(LED_ON);
lcd.createChar(0, bell);
lcd.createChar(1, note);
lcd.createChar(2, clock);
lcd.createChar(3, heart);
lcd.createChar(4, duck);
lcd.createChar(5, check);
lcd.createChar(6, cross);
lcd.createChar(7, retarrow);
lcd.home();
lcd.setCursor(0, 0);
for(int i = 0;i < 20; i++) lcd.printByte(6);
lcd.setCursor(0, 1);
lcd.printByte(6);
lcd.print(" Hello world ");
lcd.printByte(6);
lcd.setCursor(0, 2);
lcd.printByte(6);
lcd.print(" i ");
lcd.printByte(3);
lcd.print(" arduinos! ");
lcd.printByte(6);
lcd.setCursor(0, 3);
for(int i = 0;i < 20; i++) lcd.printByte(6);
// lcd.clear();
delay(1000);
lcd.setCursor(2, 1);
lcd.print(" ");
lcd.setCursor(2, 2);
lcd.print(" ");
}
void loop()
{
delay(1000);
Serial.println(second());
lcd.setCursor ( 9, 1 );
lcd.print(second());
}