How can a Java variable be different from itself?

One simple way is to use Float.NaN:

float x = Float.NaN;  // <--

if (x == x) {
    System.out.println("Ok");
} else {
    System.out.println("Not ok");
}

Not ok

You can do the same with Double.NaN.

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From JLS §15.21.1. Numerical Equality Operators == and !=:

Floating-point equality testing is performed in accordance with the rules of the IEEE 754 standard:

• If either operand is NaN, then the result of == is false but the result of != is true.

  Indeed, the test x!=x is true if and only if the value of x is NaN.

...

By the Java Language Specifications NaN is not equal to NaN.

Therefore any line that caused x to be equal to NaN would cause this, such as

double x=Math.sqrt(-1);

From the Java Language Specifications:

Floating-point operators produce no exceptions (§11). An operation that overflows produces a signed infinity, an operation that underflows produces a denormalized value or a signed zero, and an operation that has no mathematically definite result produces NaN. All numeric operations with NaN as an operand produce NaN as a result. As has already been described, NaN is unordered, so a numeric comparison operation involving one or two NaNs returns false and any != comparison involving NaN returns true, including x!=x when x is NaN.

int x = 0;
if (x == x) {
    System.out.println("Not ok");
} else {
    System.out.println("Ok");
}