TLDR; We don't really know.
WebMD has a good summary of the current state of science on this topic:
One of the most publicized and controversial theories concerns aluminum, which became a suspect in Alzheimer's disease when researchers found traces of this metal in the brains of patients with Alzheimer's disease. Many studies since then have either not been able to confirm this finding or have had questionable results.
The idea that aluminum is correlated to Alzheimer's disease came about when autopsies found elevated amounts of aluminum in the brains of diseased Alzheimer's patients. As as demonstrated in this study)
Five of 6 cerebral biopsies from patients with Alzheimer's disease also had elevated aluminium content. In 2 additional Alzheimer's brains with neurofibrillary degeneration restricted to certain anatomical areas, elevated aluminium content was found to be associated with neurofibrillary degeneration and not with senile plaques. Furthermore, elevated aluminium content in multiple cortical regions was not found in 2 vascular dementias of the elderly. While the cytotoxic concentration for human neurons is unknown, the cytotoxic concentration for cat's cerebral neurons appears to lie between 4 and 6 mug/g dry weight.
However, critics point out that the elevated aluminum levels did not occur in all cases, suggesting that, even if aluminum is correlated, it cannot be blamed as a single cause.
Other studies have failed to show a correlation (although that might be expected if aluminum is not the only cause). This study, for instance, found no statistical correlation:
There were no statistically significant differences in brain aluminum concentration between the 10 patients with Alzheimer disease, and the 9 nonneurological controls.
Studies continue along these lines, and many studies do seem to find a correlation, but since it's difficult to do direct tests on such a matter on living human brains, proxy tests must be done (on animals, or other experimental areas). This of course, leads to criticism.
Some such studies:
And of course, there are accusations of a cover-up by government and/or the aluminum industry.