Usually newspapers websites will use cookies if they want to let you read a limited number of articles. You can deleted your cookies from the telegraph or if you are using a browsers like Google Chrome or Opera with an “incognito” (google) or “private” (Opera) browsing mode (that ignores and doesn’t store cookies or history) you can try opening the link in that. For the NYtimes the later method worked for me.
I was also told that you can go to settings on your browser. Hit privacy and security. Hit delete or clear cookies.