One cannot fulfill the obligation of Netilat Yadayim in the morning by dipping his hands in a bowl of water, even if he dips both hands three times. The reason, quite simply, is that the moment his hands enter the water, they contaminate it, as the impurity on the hands is imparted into the water. Hence, this water can no longer be used for the second and third washings. According to the teachings of Kabbalah, dipping one's hands three times in a bowl of water is entirely effective, though according to the strict Halacha, this method suffices to allow one to pray. Therefore, if a person has no other alternative, he may perform the morning hand-washing by immersing his hands into a utensil of water. However, he should not recite the Beracha of "Al Netilat Yadayim" over this washing, as it is uncertain whether he fulfills the Misva.
The Shulhan Aruch writes that it is uncertain whether one fulfills the Misva by immersing his hands in three different bowls of water. Hacham David Yosef, in Halacha Berura, comments that according to the Zohar, the impurity is removed from the hands only if one washes them three times in alternating fashion, and so immersing them in the same utensil will be ineffective. Therefore, in this instance, too, if one has no alternative, he may wash his hands in this fashion – immersing them in three different bowls of water – but without reciting a Beracha.
If one was compelled to wash in this fashion – by dipping into a bowl or several bowls of water – and soon thereafter is able to wash the normal way, by pouring from a utensil, then he should wash in the usual manner, with a Beracha. If a considerable amount of time has passed since he dipped his hands in a bowl of water, then he should wash without reciting a Beracha.