Augustus saw that the wealthy aristocratic families were having declining birthrates, so he made a controversial set of social laws encouraging childbirth. (the lex Julia http://www.ancient.eu.com/article/116/)

But as for your initial question, Augustus never had a biological son. Tiberius, his adopted successor, had a son, Drusus, who died before his father (maybe murdered by Sejanus, according to Tacitus) Caligula reigned from age 25-29 and had various spats with his wives, some of whom he executed. Claudius, his successor, had a son, Brittanicus, who died young (maybe murdered by Nero, again according to Tacitus) Nero allegedly killed his pregnant wife, Poppea, and committed suicide while still a young man. Vespasian was succeeded by his son, Titus, who died after 2 years and was succeeded by his younger brother, Domitian. Domitian never had a son, and was ultimately succeeded by the 5 'good' emperors, who all coincidentally didn't have sons, until Marcus Aurelius.

So, for the first 13 emperors, starting with Augustus, only 4 had sons who were able to succeed them. While there were the obvious factors of chance of coincidence, I would agree that it is surprising that so few were able to have their biological heirs succeed them. It is important to note, however, that the Romans saw adult adoption as a totally acceptable and legitimate way to continue the family line, much like it is today in Japan.

If I could expand this a bit, why did elite families during the Imperial period also have so few children? Did it have anything to do with inheritance laws?