What if pilot feels too tired before his flight?
So if you are well rested and not ill, but still feeling tired, there has to be something else wrong. From your description, it appears to be fatigue. The common signs for fatigue are:


Falling asleep
Yawning
Poor visual acuity
Feeling "sluggish" or "drowsy"
Decreased reaction time
Decreased concentration


Although, the most commonly known reason for fatigue is lack of sleep, but any or all of the following can cause fatigue:


Lack of quality sleep
Sleep disturbances
Interruption of circadian rhythm
Mental or emotional stress
Physical exertion, such as heavy exercise
Poor health, including dehydration or poor diet


Pilots can assess their own fitness to fly by the IMSAFE mnemonic. FAA has a small handout about fatigue.

Now what pilots (and generalizing, people) can do and actually do are two very different aspects. A survey about European pilots mention:


  ... many are afraid their fatigue reports could have negative
  consequences for their professional future (i.e. reprisals by
  management) ...


FAA has a new rule which states:


  The FAA expects pilots and airlines to take joint responsibility when
  considering if a pilot is fit for duty, including fatigue resulting
  from pre-duty activities such as commuting.  At the beginning of each
  flight segment, a pilot is required to affirmatively state his or her
  fitness for duty.  If a pilot reports he or she is fatigued and unfit
  for duty, the airline must remove that pilot from duty immediately.


The answer to your question is: Yes, they can!