**Not an answer, but way too long for a comment:**

According to Ilyashenko ("Centennial history of Hilbert's 16th problem," http://www.ams.org/journals/bull/2002-39-03/S0273-0979-02-00946-1/S0273-0979-02-00946-1.pdf), the claimed result of Petrovski and Landis was disproved by Ilyashenko and Novikov (pg. 303). A citation to this disproof is not given, Ilyashenko's bibliography lists no articles by Ilyashenko & Novikov, and I can't find a paper by Ilyashenko and Novikov on the subject, so I don't believe they ever published their argument; however, Ilyashenko does point out that Landis and Petrovski claimed $H(2)=3$, but Shi Song Lin and Chen & Wang independently constructed quadratic vector fields with four limit cycles, showing that $H(2)\ge 4$. Their methods were based on Poincare Bendixon theorem on $S^{2}$ and the order of weak focus of quadratic system.

This seems annoyingly hard to find! Landis and Petrovski's paper "On the number of limit cycles of the equation ${dy\over dx}={P(x, y)\over Q(x, y)}$, where $P$ and $Q$ are polynomials of 2nd degree" is freely available online (http://www.mathnet.ru/links/b7f25c4ee0acc2e5f39e5614ea6e4c54/sm5216.pdf); unfortunately, it seems to only be available in Russian, which I can't read. Additionally, I can't find a copy of Chen & Wang's paper or of Shi's paper online (although the abstract of the latter is available), which I would suspect say at least a little bit about Landis and Petrovski's arguments.

On the plus side, explicit constructions of quadratic vector fields with four limit cycles - even really nice ones! - *are* available in English: e.g., http://arxiv.org/pdf/1002.1055v1.pdf. I can't seem to find the *simplest* examples, though, which would presumably be the nicest for trying to figure out (without access to the paper itself) what the error was.

As a side question,

Ilyashenko says he and Novikov found the error "in the early 60s" (pg. 303). Backing this up, Shi's abstract mentions that the question "Is there a quadratic vector field with exactly 4 limit cycles?" was asked at a 1974 symposium on Hilbert's problems, so presumably the error was already known (Landis & Petrovski published in 1955). Then things get a bit tricky for me to track down: presumably, the symposium in question was "Mathematical developments arising from Hilbert's problems" (http://www.ams.org/bookstore-getitem/item=PSPUM-28); however, the table of contents for the proceedings of that symposium show no talk or paper on the 16th problem. Thus, although I'd imagine more specific discussion of when the error was noticed (and maybe even what it was!) would have happened at this symposium, I can't find it.

Beyond that, I have no idea when exactly the error was discovered.