If in doubt then remove the other person.
E.g., "I went to the park" - correct. "Me went to the park" - incorrect. Therefore "Jane and I went to the park" - correct.
"It was me" - correct. "It was I" - incorrect. Therefore "It was Jane and me" - correct.
It only sounds wrong because it became a trend for people to say "and I" at ALL times, for fear of getting it wrong.
Slightly on a tangent, but it's a bit like the way words such as 'serviette' came into usage, and (in England) are traditionally deemed 'lower class' terms. The masses adopted certain words in attempts to sound 'higher class' than they were, without realising that the upper-middle and upper classes used the more lowly sounding 'napkin.' Therefore 'serviette' (along with many other words) became a giveaway word that someone was purporting to be of 'better breeding' than they really were.
Sorry to go off-topic a bit - I just find it fascinating. I guess my tenuous link was that 'serviette' was a word that people thought sounded right, but didn't achieve the effect they were after - a bit like the 'and I' dilemma!