The New York Times has a good piece saying that, in actuality, it succeeded, in at least driving poverty rates down a bit, and surely, in keeping them from expanding. It adds that, to the degree the War on Poverty is considered a failure by some, that's because the right wing has long attempted to strangle the child in its crib.
But, could it have done better?
Yes, certainly in reducing housing segregation, one of LBJ's biggest goals in the last couple of years of his administration.
ProPublica documented a year ago, since LBJ signed into law the 1968 Fair Housing Act, how one of its requirements, not distributing block grant money from the Department of Housing and Urban Development to communities who did not appear to have plans to use said grants in a non-discriminatory manner, was gutted by Tricky Dick Nixon. HUD Secretary George Romney continued to push to fulfill the letter and spirit of the act, though, and for his pains, got fired.
Since then, ProPublica noted, both GOP and Democratic HUD administrations have largely only given lip service to this.
Results? Continued segregation in public school districts, variable access to urban and suburban amenities, etc.
That said, and to give a kudo to the Obama Administration, in the past year, things ticked up a bit, ProPublica says in a follow-up.
Do read the original main piece, though. LBJ knew just what was at stake when he pushed so hard for this. So did his opponents.