One expects the Brauer-Manin obstruction to control rational points on 1-parameter families of conics and quadrics over a number field when the base curve has genus 0. Results in this direction have recently been obtained as a consequence of progress in additive combinatorics. On the other hand, it is easy to construct a family of 2-dimensional quadrics over a curve with just one rational point over Q, which is a counterexample to the Hasse principle not detected by the Brauer-Manin obstruction or its more sophisticated version. Conic bundles with similar properties exist over real quadratic fields (but most certainly not over Q).