Stripping out all /* */ comments from a paragraph
You didn't specify whether or not nested comments are allowed, but your code trims only the shorter comment. For example, case 4 would yield Text  Text with nesting but Text  */ Text without. It would be helpful to include some unit tests to demonstrate the intent.

Your solution is pretty straight-forward with only one real complication. I would remove the moveAheadBy incrementer and instead perform an extra increment where you're setting moveAheadBy to 2. It's minor, but it removes unnecessary state.

To help with the readability, introduce a variable for the length of the input since it's used in so many places.

Another solution would be to use indexOf to seek ahead to the next open/close. The algorithm will still be \$O(n)\$, but the fewer number of append calls could add up with very, very long strings. This is more to show an alternate method than to improve on yours.

public static String stripComments(final String input) {
    StringBuilder buf = new StringBuilder();
    int current = 0;
    int end = input.length();

    while (current &lt; end) {
        int nextOpen = input.indexOf("/*", current);
        if (nextOpen == -1) {
            // no more comments
            buf.append(input.substring(current));
            break;
        }
        buf.append(input.substring(current, nextOpen));
        int nextClose = input.indexOf("*/", nextOpen + 2);
        if (nextClose == -1) {
            // final comment not closed
            break;
        }
        current = nextClose + 2;
    }

    return buf.toString();
}


Unit Tests

Now that the method is public, let's add some tests.

@Test
public void testNone() {
    assertThat(stripComments("No comments at all"), is("No comments at all"));
}

@Test
public void testSingle() {
    assertThat(stripComments("Single /* foo */ comment"), is("Single  comment"));
}

@Test
public void testDisjoint() {
    assertThat(stripComments("Two /* foo */ disjoint /* bar */ comments"), is("Two  disjoint  comments"));
}

@Test
public void testNested() {
    assertThat(stripComments("Two /* foo /* nested */ baz */ comments"), is("Two  baz */ comments"));
}

@Test
public void testUnclosed() {
    assertThat(stripComments("Unclosed /* foo comment"), is("Unclosed "));
}

@Test
public void testUnopened() {
    assertThat(stripComments("Unopened */ comment"), is("Unopened */ comment"));
}