Burke's Extinct tells the sad story of how Sir Thomas Burgh jr married Elizabeth Owen, but she bestowed her affections on another. Her kids weren't his, and were declared bastards by a Parliament with nothing better to do.
We're then told that Sir Thomas married Alice and had some kids of his own, including his heir William. The book goes on to say that William's son Thomas was summoned to Parliament from 1563.
Other sources disagree about the 2nd marriage. The father, Thomas sr, died in 1550 and his successor William is said to be his younger son, not his grandson. This line is followed by Cokayne
https://dcms.lds.org/delivery/DeliveryManagerServlet?dps_pid=IE83371 (Big PDF download)
This William married Katherine Clinton and had Henry (killed in a duel), Thomas (heir) and John. John has a memorial in Westminster Abbey, in very non-Tudor language, which says who his parents were and says he died in 1594 at age 52 ("fighting the enemy", though really it was another duel.)
This would imply that Thomas was born before 1542, which fits with the writs of summons from Jan 1563.
The only fly in this ointment is that their mother's admon was dated 1622. But sometimes these things could be delayed.
But by the time Vicary Gibbs was doing the 2nd edn, it was known that there wasn't any delay - the lady was buried in 1621.
Which makes it unlikely that she was having sons in 1541-42. However, the alleged writs to Thomas from 1563 were bogus. Dugdale had evidently spotted there was something wrong, and supposed that they must have been sent to William really, but he was miscalled Thomas. In fact William lived until 1584 and Thomas was only 26 then.
So now we're left with the puzzle of how William's heir Thomas born around 1557 has a younger brother John born in 1542.
Anybody got the Complete Peerage 2nd edn additions and corrections volume?