I have also heard that our Jewish ancestors, specifically during the time of the Temple, used to wear tefillin throughout the day.
Looking at the Kitzer Shulchan Aruch, as well as related questions, gives a good idea as to why the idea of wearing tefillin throughout the day is possible but not overtly practical.
Kitzer Siman 10:1
"Whoever is scrupulous in the mitzvah of tefillin, to treat them [in accordance] with [their] holiness, not to speak idle words or of mundane matters while wearing them, he will live a long life and is assured that he is destined for the World to Come."
Similarly, Siman 10:16
"The entire time that the tefillin are on a person, he must not divert his mind from them at all, except for the time when he is praying the Shemoneh Esrei, or when he is learning Torah."
"It is forbidden to eat a meal while wearing tefillin, but it is permitted to partake of a snack while wearing them. With regard to sleeping, it is forbidden to take even a short nap while wearing them."
Significantly (Siman 10:20):
"Because [in truth the optimal form of] the mitzvah [is for] the tefillin to be worn the entire day; [and it is] only because it is difficult to maintain a clean body (see Siman 10:23) [that] one removes them immediately after the Shacharis prayer."
Siman 10:23 :
"[The wearing of] tefillin requires [that one maintain] a clean body and one must [therefore] be careful not to pass gas while wearing them."
Note that "Eating up to the volume of an egg is considered a snack with regard to this halacha" (Mishnah Berurah 40:20)."
And that "Sleeping while wearing tefillin is prohibited lest one inadvertently pass gas while sleeping (Mishnah Berurah 44:1). According to some authorities, it is permitted to doze briefly (by placing one's head between his knees - see Beur HaGra ibid. 3 and Magen Avraham ibid. 3) up to the amount of time that it takes to walk 100 amos. This is approximately one sixty-seventh of an hour [less than a minute] (see Mishnah Berurah 44:4)."
See All day tefillin
It seems that the main reasons to not prolong the use of tefillin center around uncleanliness, both in thought and in physical action. However, as noted, the optimal form of this mitzvah is to be worn throughout the entire day. I recommend trying to wear tefillin for a prolonged period as a spiritual endeavor. Just make sure to remove them when you eat anything more than the size of an egg, need to use the restroom, or take a nap.