In the 1800s in the Western US, there were a number of telegraph lines running through territory mostly inhabited by Indians. Did they ever sabotage the lines?
From what I've been able to dig up, the answer appears to be yes, but not as much as you'd think.
It appears the Telegraph companies saw the danger every bit as clearly as you did, and actively took steps to prevent it. They made sure to meet with the chiefs through whose territory they ran lines, hired them to help construct the lines, and generally took pains to treat them with far more respect and deference than white men typically showed Native Americans.
The importance of having a good understanding and keeping on friendly terms with the Indians was well understood, and everything was done, both then and during the period of the construction of the line, to prevent the occurrence of anything that would lead to trouble with them. ...
...That this good feeling with the Indians was maintained throughout, was also in a measure due to a general order issued at the start, that any man of the expedition getting into trouble with the Indians, or their squaws, would be immediately dismissed from the service, and this rule was strictly enforced.
Apparently all this effort did pay off.
In connection with our treatment of the Indians during the period of this work, it might be well for me to mention that the consideration we manifested toward them appeared, in after years, to be fully appreciated. This was instanced in 1863, two years after the completion of the overland telegraph line, when an Indian war broke out on the overland route, ...
During all these troubles, the telegraph line was not disturbed, and, if my recollection serves me right, no stage station in which a telegraph office was established was ever burned; nor was an employee of the Company ever molested or injured by the Indians. They seemed to look on the telegraph people as another tribe and against which they had no hostility.
However, there are, as you suspected, always exceptions. In fact, if the lines were being actively used against them, it would be stupid not to cut them:
Sweetwater Station, in the South Pass, was attacked by a band of Sioux Indians...The Indians promptly returned the fire, and the fight lasted for several days. At the first moment of attack the operator telegraphed to the nearest fort for troops to come to the rescue. Shortly after having done so, the wires were cut by the Indians in the hope that it would cut off communication for relief.
I'd highly suggest reading the entire link above. Fascinating stuff.