Gemara

A tradition from Moshe from Sinai teaches that Tefilin must be sewn with sinews. The same applies to Torah.

Rav: I saw R. Chiya's Tefilin. They were sewn with flax.

The Halachah does not follow R. Chiya.

Menachos 34b (Beraisa) If the Parshiyos were written on one parchment and (the Klaf was folded like an accordion, so) each Parshah was in its own Bayis, this is Kosher.

Rebbi says, there must be a space between them;

Chachamim say, there need not be.

All agree that a string must separate the Batim (compartments that house the Parshiyos). If the separation between Batim is not recognizable, it is Pasul.

35a (Rav Huna): Tefilin are Kosher as long as Tavla (the leather) is intact.

(Rav Chisda): If two are torn, they are Kosher. If three are torn, they are Pasul.

(Rava): Even when two are torn, it is Pasul if they are adjacent to each other.

They are Pasul only if the Tefilin are new.

(Rav Papa): The stitching must be square, and also the diagonals.

Rishonim

Rif and Rosh (Hilchos Tefilin 7b and 10): We disqualify Tefilin if Tavla is not intact, i.e. three are torn, when the Tefilin are old. If they are new, it is Kosher.

Rambam (Hilchos Tefilin 3:10): We sew Tefilin in a square. It is a known Halachah that there are three stitches on each side, i.e. 12 in all. If he did 10 or 14, he may do so. Every thread goes around in both directions.

Rabbeinu Manoach: This 'known Halachah' is not a Gemara and it is not in the the Rif. It is a universal custom. Shimusha Raba brings it.

Rambam (18): If two stitches snapped next to each other, or three snapped, even if they are not adjacent, it is Pasul. This refers to old Tefilin. New Tefilin are Kosher as long as Pnei Tavla lasts. It is new if one can hold part of the leather from which the stitching tore and, and it will not break. If not, it is old.

Bi'ur Halachah (33:2 Sof DH Lachush): 'Pnei Tavla' refers to the squareness of the stitching.

Shimusha Raba (Hilchos Tefilin of a Ga'on, in the Rosh after Siman 31): One must sew Tefilin with sinews, like Rav Papa taught. Some require sinews of an ox. We asked about sinews of a small animal (not used for work, i.e. sheep or goats). The question was not settled, so we are stringent. Some say that Rav Yosef permitted. We sew it with 12 stitches, aside from d'Ayil v'Nafik. Rava is Machshir if there are 10, for this is the number (of Shevatim) excluding the king (Yehudah) and servant (Levi). Rav Papa says that if there are 14, this is fine, for this is including Efrayim and Menasheh.

Rosh (Sof Hilchos Tefilin): After making the base square, one sews it with sinews of a Kosher Behemah. He makes three stitches on each side, 12 in all. The stitching goes around on two sides. He stitches between every two (adjacent) Batim. If the stitching snapped in three places, he must sew it again.

Rosh (2): All agree that a string must separate the Batim. Chachamim do not require space between them, but they agree that if the Parshiyos were written on one parchment, there must be a string between Batim, Some require a string even when they are written on four parchments. They say that the Seifa (a string must separate) refers back to the Reisha (they are written on four parchments). The words 'all agree that' connotes that it refers to the previous law that Rebbi and Chachamim argued about (when they are on one parchment).

Rosh (12): According to some texts, the Yerushalmi discusses a case in which the stitching on the base snapped. If it snapped in one place, it is Kosher. If it snapped in two places, it is Kosher, but not according to letter of the law.

Poskim

Shulchan Aruch (OC 32:51): One sews three stitches on every side. The thread must go around two sides, and he must pass a stitch between every two Batim.

Beis Yosef (DH v'?hut): The Rambam and Rosh say that the stitching goes back and forth. They learn from Shimusha Raba, which approves of 14 stitches, for there are 12 excluding d'Ayil v'Nafik. Sefer ha'Terumos and the Mordechai explain that d'Ayil v'Nafik is the first insertion of the needle. Ri Alexandri explains that each stitch must go in both directions, back and forth. He says that it must be all with one thread. If it snapped, the Tefilin are Pasul. I do not know his source.

Magen Avraham (68, citing the Bach): If the thread snapped while sewing, it is Pasul, for this shows that (it is weak and) it will snap again, but if the thread is too short, he may finish with a second thread.

Mishnah Berurah (231): The Taz holds that if it snapped while sewing, one can tie the ends. Others permit tying only if the thread was too short. If one has no other sinews, one may rely on the Taz. Below (33:2) the Shulchan Aruch discusses if it snapped after sewing.

Beis Yosef (DH v"Ya'avir): Tosfos, the Rosh and Mordechai say that even though the Gemara connotes that passing a stitch between Batim refers to the previous law, the custom is to do so even when the Parshiyos are on four parchments. The Mordechai says that this is primary. The Rambam says that the custom is to pass a sinew used for stitching between the Batim. Perhaps Shimusha Raba said 'excluding d'Ayil v'Nafik' to exclude the stitch between the Batim. The stitching ties the upper leather between the Batim with the lower leather of the base, to divide between them. Perhaps this is only when the Parshiyos are on one parchment; one should not be lenient. Also Semag is stringent; Semak is lenient when they are on four parchments. Orchos Chayim says that some are stringent to pass a stitch thread between the Batim. It seems that any thread suffices. Some say that no thread is needed. The Gemara requires one only when they are moist, lest they stick together and the crevice will not be evident.

Mishnah Berurah (228): The leather of the Batim must extend below the base, so the stitches tie it to the base. Baruch she'Omar disqualified Tefilin that did not fulfill this.

Mishnah Berurah (230): If he did not pass a stitch between the Batim, b'Di'eved it is Kosher.

Rema: If he did only 10 stitches or less, it is not disqualified.

Beis Yosef (DH u'Mah): Shimusha Raba connotes that if there are less than 10, it is Pasul. The Mordechai is Machshir. It seems that the Rambam and all the Poskim agree.

Shulchan Aruch: Some say that these 12 stitches should be with one thread.

Shulchan Aruch (33:2): If the stitches snapped, according to the Rambam, if there were two (snapped) stitches next to each other, or three stitches snapped anywhere, it is Pasul. This refers to old Tefilin. Regarding new Tefilin, as long as the base of the Batim is intact, it is Kosher.

Beis Yosef (Sof Siman 32 DH v'Im): Sefer ha'Terumos says that the text in the Yerushalmi says that the stitching snapped, and that Shimusha Raba, Bahag and the Rif disqualify if three stitches snapped. We permit only two, and only in pressed circumstances, for it is disgraceful. The Rambam explains 'Tefilin are Kosher as long as Tavla is intact... if two are torn, they are Kosher...'to discuss when the stitching tore. The Meforshim explain differently. When one cannot be stringent, he should rely on the lenient opinions rather than neglect the Mitzvah of Tefilin.

Mishnah Berurah (12): The other Poskim are always Machshir when two snapped, and always disqualify when three snapped.

Rema: Some are Machshir old Tefilin and disqualify new Tefilin if two or three threads snapped. It is good to be concerned for both opinions.