I somewhat disagree. In terms of neuron counts, the elephant brain is larger, although admittedly most of that is in the elephant's much larger cerebellum - presumably as a brute force solution to the unique control complexity of the trunk appendage.

The elephant's cortex (which seems key to human style general intelligence) is roughly 1/3 of our neuron count (I'm guessing from memory) - comparable to that of the chimp. There is some recent evidence that elephants may have sophisticated low frequency communication. They even have weird death 'burial' rituals. They can solve complex puzzles in captivity. There are even a few cases of elephants learning to speak some simple human words with their trunks.

So in short, elephants seem to have just about as much general intelligence as you'd expect given their cortical neuron count. The upper range of the elephant's raw brain capability probably comes close to the lower range of human capability.

The large apparent gap in actual adult intelligence is due to the enormous nonlinear amplification effects of human culture and education.