Can you use 2 or more OR conditions in an if statement?

You need to code your tests differently:

if (number==1 || number==2 || number==3) {
    cout << "Your number was 1, 2, or 3." << endl;
}
else if (number==4 || number==5 || number==6) {
    cout << "Your number was 4, 5, or 6." << endl;
}
else {
    cout << "Your number was above 6." << endl;
}

The way you were doing it, the first condition was being interpreted as if it were written like this

if ( (number == 1) || 2 || 3 ) {

The logical or operator (||) is defined to evaluate to a true value if the left side is true or if the left side is false and the right side is true. Since 2 is a true value (as is 3), the expression evaluates to true regardless of the value of number.

While you can (as others have shown) re-write your tests to allow what you want, I think it's also worth considering a couple of alternatives. One would be a switch statement:

switch (number) { 
    case 1:
    case 2:
    case 3:
        cout << "Your number was 1, 2, or 3." << endl;
        break;
    case 4:
    case 5:
    case 6: 
        cout << "Your number was 4, 5, or 6." << endl;
        break;
    default:
        cout << "Your number was above 6." << endl;
}

Personally, I'd probably do something like this though:

char const *msgs[] = {
    "Your number was 1, 2, or 3.\n",
    "Your number was 4, 5, or 6.\n"
};

if (number < 1 || number > 6)
    std::cout << "Your number was outside the range 1..6.\n";
else
    std::cout << msgs[(number-1)/3];

Note that as it stands right now, your code says that 0 and all negative numbers are greater than 6. I've left this alone in the first example, but fixed it in the second.