Do you think Brave New World could happen today?

Brave New World is a work of fiction based on scientific and psychological ideas that are obsolete at best. It does not describe what even Huxley would have considered a probable future and was not intended as a work of prediction but rather intended as a cautionary tale about certain social ideas and trends. It glamorizes the primitive and underestimates the potential of automation. It doesn't make sense to ask if a completely imaginary world intended as the setting for a piece of fiction could "happen today." Instead, we could think about whether some of the trends that Huxley was criticizing are still in evidence and whether those trends are likely to persist or not.
The first worrisome trend that Huxley did get right is the growth of inequality, with an educated elite enjoying the advantages of technological progress and people who are less educated having fewer opportunities. As genetic engineering becomes more sophisticated, one can imagine a class divide widening between those who can afford genetically engineered offspring and those who cannot. We also do seem to be seeing an increase in assortative mating, which has a similar effect. 
What Huxley underestimates is developments in robotics and artificial intelligence. Increasingly, simple manual tasks will be done by machines, not a subclass of humans.
As for mindless entertainment and psychoactive chemicals substituting for human interactions and a hook-up culture taking the place of long term relationships, that may indeed be happening among some groups, but it is hard to predict whether it will be universal or in any way encouraged by the state.