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
...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.