I've worked on lots of these and have only seen one of the infamous bad HV transformers. Normally HV failures are due to a semiconductor or capacitor failure. I must mention that the HV unit can put out very weak X-ray radiation and and creates lots of ions in the air but both are in sufficiently small quantities that there is little risk of injury. Nonetheless, many people feel a tad queasy during or after working near high voltage. Like anything else, it gets less noticeable with successive exposures.
Many people are quite worried the first time they experience the sensation but it is harmless.
Please note that I am not a doctor nor a radiation specialist so refer back to the CYA clause at the beginning of this document. Certain unlikely faults that I have never run across could cause large amounts of X-rays to be produced which would definitely be harmful. Connect a clip lead between the chassis and the shaft of a long narrow plastic-handled screwdriver. Work the end of the screwdriver under the big suction cup on the top of the tube until you hit metal. Just go slowly and use only one hand.