I'm not particularly well known, but I've been in fairly visible positions in companies large and small for most of my career, so inevitably I get "invitations" on LinkedIn pretty regularly.
I'm pretty liberal about accepting these - after all LinkedIn is all about maintaining a sound professional network. Anyone I've ever worked with, or even who I could have worked with but maybe never got to know, is fine. It's not like Facebook, which I limit to real-life friends and acquaintances - maybe I haven't seen them in a very long while, but at some point I could have gone for a beer or a coffee with them, and I still would happily if the opportunity arose.
But actually most of these "invitations" come from people who not only I've never heard of, but even after I look at their profile I have no idea why they've suddenly decided to get in touch with me. Maybe they think I might give them a job or put some work their way, maybe they have some other reason.
Yet they invariably come with the default LinkedIn text "I'd like to connect with you on LinkedIn". I'm sure you would, otherwise you wouldn't have sent me an invite, but why? Wouldn't it be nice to take the time to say, "I see you were at XXX a few years ago and so was I, right now I'm looking around for a new opportunity and I thought you might be able to help".
Or, "My company makes widgetators and from your role, it looks as though you might sometimes need them."
Or, "I'm a recruiter and I like to have thousands of people in my LinkedIn contact list even though you'll probably never hear from me." (I get quite a few of those, if you have "recruiter" in your title or role you can reasonably presume that I will not accept your invite unless I know you personally).
Or even, "You have no idea who I am and to be honest I have no idea who you are either, but I like to have a huge list of contacts and I pulled your name from a hat today so I thought I'd send you this invite".
Sending the default invite text is not only lazy and impolite, it's also self defeating. If I've never heard of you and you give me absolutely no reason why you want me to add you to my contact list, what on earth makes you think I'm going to accept it?