The answer is that, although there seems to be an association there is no clear scientific proof for it.
Source: Adult and childhood leukemia near a high-power radio station in Rome, Italy. - Michelozzi, Am J Epidemiol, 2002 (bold is mine)
In the 10-km area around the station, with 49,656 residents (in 1991),
leukemia mortality among adults (aged >14 years; 40 cases) in
1987-1998 and childhood leukemia incidence (eight cases) in 1987-1999
were evaluated. The risk of childhood leukemia was higher than
expected for the distance up to 6 km from the radio station
(standardized incidence rate = 2.2, 95% confidence interval: 1.0,
4.1), and there was a significant decline in risk with increasing distance both for male mortality (p = 0.03) and for childhood leukemia
(p = 0.036). The study has limitations because of the small number of
cases and the lack of exposure data. Although the study adds evidence
of an excess of leukemia in a population living near high-power radio
transmitters, no causal implication can be drawn. There is still
insufficient scientific knowledge, and new epidemiologic studies are
needed to clarify a possible leukemogenic effect of residential
exposure to radio frequency radiation.
Aside from the specific Radio Vaticana's case, there are other studies that analyze similar issues. The problems, however, are always the same: low power studies, with a huge amount of counfounding variables (problems that are generally noted in the studies themselves).
Investigation of increased incidence in childhood leukemia near radio towers in Hawaii: preliminary observations - Maskarinek et al., J Environ Pathol Toxicol Oncol, 1994
The clustering may have been a chance event, but because of its peculiar characteristics, we feel it should be noted.
Cancer incidence and mortality and proximity to TV towers. - Hocking et al, Med J Aust, 1996
The calculated levels of RFR in the areas with increased childhood leukaemia incidence and mortality are substantially below the current Australian public safety standard. More detailed studies (e.g., relating cases to power density contours) are required to replicate any association and to look for dose-response relationships before any conclusions can be drawn.
Cancer Incidence near Radio and Television Transmitters in Great Britain I. Sutton Coldfield Transmitter - Dolk et al., Am J Epidemiol, 1997
A small area study of cancer incidence in 1974-1986 was carried out to investigate an unconfirmed report of a "cluster" of leukemias and lymphomas near the Sutton Coldfield television (TV) and frequency modulation (FM) radio transmitter in the West Midlands, England
In conclusion, the results of this study confirm that there was an excess of adult leukemia within the vicinity of the Sutton Coldfield TV/FM transmitter in the period 1974-1986, accompanied by a decline in risk with distance from the transmitter. Further monitoring of cancer statistics in the area appears warranted. No causal implications regarding radio and TV transmitters can be drawn from this finding, based as it is on a single "cluster" investigation. Results of a study of cancer incidence around all other high power radio and TV transmitters in Great Britain are given in the accompanying paper in order to put the present results in wider context.
The accompanying paper is:
Cancer Incidence near Radio and Television Transmitters in Great Britain II. All High Power Transmitters - Dolk et al., Am J Epidemiol, 1997
We report here findings for adult leukemias, skin melanoma, and bladder cancer near the other 20 high power radio and TV transmitters in Great Britain. Eight of these transmit FM radio frequencies and three transmit TV frequencies at power equivalent to Sutton Coldfield but none transmit at exactly the same combination of frequencies and power as Sutton Coldfield
For childhood leukemia and brain cancer, and adult skin melanoma and bladder cancer, results were not indicative of a decline in risk with distance from transmitters.
The magnitude and pattern of risk found in the Sutton Coldfield study did not appear to be replicated. The authors conclude that the results at most give no more than very weak support to the Sutton Coldfield findings.