Is it possible to do strict mocks with Mockito?

You could use verifyNoMoreInteractions. It's useful if the tested class catches exceptions.

@Test
public void testVerifyNoMoreInteractions() throws Exception {
    final MyInterface mock = Mockito.mock(MyInterface.class);

    new MyObject().doSomething(mock);

    verifyNoMoreInteractions(mock); // throws exception
}

private static class MyObject {
    public void doSomething(final MyInterface myInterface) {
        try {
            myInterface.doSomethingElse();
        } catch (Exception e) {
            // ignored
        }
    }
}

private static interface MyInterface {
    void doSomethingElse();
}

Result:

org.mockito.exceptions.verification.NoInteractionsWanted: 
No interactions wanted here:
-> at hu.palacsint.CatchTest.testVerifyNoMoreInteractions(CatchTest.java:18)
But found this interaction:
-> at hu.palacsint.CatchTest$MyObject.doSomething(CatchTest.java:24)
Actually, above is the only interaction with this mock.
    at hu.palacsint.stackoverflow.y2013.q8003278.CatchTest.testVerifyNoMoreInteractions(CatchTest.java:18)
    at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
    at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:57)
    at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
    at java.lang.reflect.Method.invoke(Method.java:601)
    at org.junit.runners.model.FrameworkMethod$1.runReflectiveCall(FrameworkMethod.java:44)
    ...

Add this @Rule to your test class as a public field:

@RunWith(JUnitParamsRunner.class)
public class MyClassTests {

    @Rule
    public MockitoRule mockito = MockitoJUnit.rule().strictness(Strictness.STRICT_STUBS);

    @Test
    ....
}

This value was added to Mockito in version 2.3.0

From the documentation:

Ensures clean tests, reduces test code duplication, improves debuggability. Offers best combination of flexibility and productivity. Highly recommended. Planned as default for Mockito v3. Adds following behavior:

• Improved productivity: the test fails early when code under test invokes stubbed method with different arguments (see PotentialStubbingProblem).
• Cleaner tests without unnecessary stubbings: the test fails when unused stubs are present (see UnnecessaryStubbingException).
• Cleaner, more DRY tests ("Don't Repeat Yourself"): If you use Mockito.verifyNoMoreInteractions(Object...) you no longer need to explicitly verify stubbed invocations. They are automatically verified for you.

What do you want it to do?

You can set it to RETURN_SMART_NULLS, which avoids the NPE and includes some useful info.

You could replace this with a custom implementation, for example, that throws an exception from its answer method:

@Test
public void test() {
    Object mock = Mockito.mock(Object.class, new NullPointerExceptionAnswer());
    String s = mock.toString(); // Breaks here, as intended.
    assertEquals("", s);
}

class NullPointerExceptionAnswer<T> implements Answer<T> {
    @Override
    public T answer(InvocationOnMock invocation) throws Throwable {
        throw new NullPointerException();
    }
}