For the purpose of this discussion, we'll think of antiquity as the time significantly before Christ. In fact, discoveries of ancient objects made near the time of Christ continue to guide our perspective on history.
It's difficult to say exactly when the first tear bottles came into being, however, in the Old Testament of the Bible (KJV), a reference to collecting tears in a bottle appears in Psalm 56:8 when David prays to God, “Thou tellest my wanderings, put thou my tears in Thy bottle; are they not in Thy Book?” David lived from 1055-1015 B.C. and wrote Psalm 56 about 1020 B.C. Tear bottles of the time might be made of glass, pottery, for sardonyx stone. Wineskins or animal skins were also a common vessel for carrying fluids.
One might speculate that tear bottles were common enough during these times that David would make reference to the concept of collecting tears in a bottle so that his audiences would understand his message. Perhaps not. The Psalm reference may have purely metaphorical, only to inspire later use of tear bottles. Interestingly, tear bottles dating from 100 A.D. are still in existence today and are occasionally sold by antiquities dealers. However, I've yet to photograph tear bottles that date earlier than about 100 A.D.
When Middle Eastern tomb raiders around 100 A.D. found small ceramic bottles in tombs of the wealthy, they believed them to be lachrymae. It was common for nobility to be buried with precious items, sometimes gold, but often jewelry. Middle class citizens were occasionally buried with pots and pans.
Because so many small bottles were found in tombs, the theory was developed that they were part of the mourning ritual. The theory was that mourners would cry into the bottles as a sign of respect. It was also been held that mourners would be paid to attend the funerals of the wealthy: filling tear bottles and wailing loudly to create dramatic impact.
The items called tear bottles were rare in most burials, but common among the wealthy. Egyptian pharaohs were buried with hundreds of these small decorative items. It is still unclear if they developed the theory of tear bottles or lachrymatory out of the blue, or if there was historical precedent.
Many scholars today believe the ancient bottles were used for perfumes or medicinal oils (both considered valuable and necessary in the afterlife). Size in an important indication of the use. Many bottles dated to the Hellenistic Period (c. 300 B.C.) are quite larges (11-25 cm tall) and would not be practical tear bottles. The only way to know for sure how the bottles were used would be to test the residue in historic bottles to determine if they were used for tears or fragrances. This will certainly happen eventually, but until then, each of us can chose our own belief.