Case 1:

void ThreadTimesClosure::do_unlocked() {
  EXCEPTION_MARK;
  for (int i = 0; i < _count; i++) {
    Handle s = java_lang_String::create_from_str(_names_chars[i],  CHECK);
    _names_strings->obj_at_put(i, s());
  }
}

create_from_str can throw OOME as per the use of CHECK macro, but the EXCEPTION_MARK would cause us to crash if this happened. The EXCEPTION_MARK is not wanted here.

case 2:

void ThreadTimesClosure::do_thread(Thread* thread) {
 ...
  EXCEPTION_MARK;
  ResourceMark rm(THREAD); // thread->name() uses ResourceArea

  assert(thread->name() != NULL, "All threads should have a name");
  _names_chars[_count] = os::strdup(thread->name());
  _times->long_at_put(_count, os::is_thread_cpu_time_supported() ?
                        os::thread_cpu_time(thread) : -1);
  _count++;
}

This EXCEPTION_MARK is superfluous as no code raises exceptions.

If the code wants to assert it is not called with an exception pending then it should assert that directly rather than using the EXCEPTION_MARK.