If the primary (120V) windings are driven such that they all tie to a single common point, and the secondary windings (HV) are connected in series, you will break down the insulation between the secondary winding and the core laminations of the transformer.
When the transformer was built, it was subjected to what is called a HiPot test, were a high voltage is applied between secondary-primary, secondary-core, primary-core until the insulation breaks down and an arc-over occurs. The spec may be something like 5KV for the secondary to anything else, and 2KV from primary to core.
If you connect the secondaries in-series, the end transformers secondary to core breakdown voltage will be exceed by two or three times what they have ever been tested to. Very likely, you will have a arc-over and a huge hazard.
Even if you cheat and try to mount the cores so that they are electrically floating (instead of being bolted to a metallic chassis), the primaries are still tied to common point, so the arc-over will just move to the primary windings.