I will pay you $1,000,000 to fix my problem.... No seriously.

Sounds like a ground loop to me. Here's what may be going on.

Telephone cables are powered by the central office to which they are connected, typically at -48 volts DC (meaning that the ground is at higher electrical potential to prevent corrosion). The voice and data signals ride on top of this bias. This is what allows telephones to operate without power at the premises, such as in the event of a power failure.

Each connection is formed by a pair of cables which form a loop (residential houses are typically serviced with two such pairs). Each pair is of cables is twisted with a spacing between turns to cancel out the electrical and magnetic interference caused by the transmitted signal.

Coaxial cables on the other hand are not powered and are not twisted. A cable connection consists of a single conductor which connects point A to point B. There is also a woven shield that surrounds the conductor to eliminate noise.

NOTE: Before cable internet became widespread most cable installations used cheap cable, typically RG-59 cable. This cable grade is fine for analog television, but is not suitable for digital cable, satellite or internet. It must be replaced with cable that has solid conductors and full shielding, typically RG-6 cable. Make sure that this has been done on premises; it is your responsibility inside of your house, but your ISPs responsibility beyond the demarcation point. If they replaced the cable outside, it's probably okay.

Moving on,

Since coaxial cables are used for signal transmission only (they do not deliver power), the transceivers on each end must provide their own, typically from the premises' AC main, but battery power may be used in the event of a power failure (cable voice modems typically have an internal battery). Since each end is powered individually without a common ground between them the transceivers must be careful to avoid unintended current flow on the conductor as this will cause the signal to become unintelligible. The transceivers themselves are designed to compensate for this, but there's one more piece of the puzzle that I mentioned above...

Coaxial cables are shielded. They're completely surrounded by a metal shield across the whole length of the wire, enclosing the conductor in a Faraday cage. The shield wire itself must be grounded and unpowered, otherwise the shield will not remain electrically neutral which would impart a current on the conductor contained within. At a bare minimum, the shield should be grounded at the ISP's termination point (usually a local node, but sometimes a CO). Ideally, it would be grounded at both ends, but this can pose a problem. It is often assumed that ground = ground = ground, however this is not true. Earth grounds are simply the most ideal reference points that we have available, and there can be a potential difference between two earth ground points that are some physical distance apart, or have no low impedance connection between them. For example, it's common to use a municipal water main as an earth ground, but if the local soil is particularly dry (ground spike, typically driven 9-12 feet into the ground), there may be a potential difference between the local water table and the municipal water supply. Another example, there may be a potential difference between ground points in Los Angeles and New York.

If this is the case, then current may flow across the shield, which will destroy the quality of the signal that it's trying to protect and possibly cause the transceivers to seize up. You should find out how your cable's shield is grounded. If it's grounded to the AC box (this is typically what most cable installers do), try switching it over to the municipal water supply or even disconnecting it all together.