I have recently exchanged business cards from several different people from several different companies in several different contexts. I have gone into LinkedIn and added them as a connection. Which they then reject. This seems very strange to me.
Would you give someone your business card and then reject their LinkedIn invite? What's the difference between a LinkedIn connection and a business card? Where does LinkedIn and exchanging business cards diverge in the way you use them?
Alternative
Rather than answering your question, I think it would be better to suggest an alternative that resolves the issue. The next time you exchange business cards, ask the recipient, "Can I add you as a connection on LinkedIn?" Then invite or not, accordingly.
Alternative #2
I agree with DONM, but a question. Do you personalise the request, or just press 'connect'? I try to always draft a bit of specific info from the discussion; which means you do it as soon as obviously. Thank them, comment on the topic you discussed or the event you attended. The request to link I then almost add as an after thought.
Sure, some folk don't connect, but not common.
J