After people had stopped asking "isn't it a little early for a new entry in the main Battlefield series?" they started to ask "why didn't you make Bad Company 3?" It does seem that by slapping the Bad Company logo on the front, the developers could do almost everything they wanted to with Battlefield 4 without the risk of over-using the original franchise. Taking on Reddit, Ian Tornay explained that there's a lot of love for Bad Company at DICE, and that people are still proud with their work on the series.
"I can tell you that it's not due to any lack of love for BC inside of DICE," he wrote. "Everyone I've talked to there is very proud and fond of it."
"[I] would be very interested to nail down for myself what it is about BC that is special to everyone and how we can incorporate that magic."
That's still not an explanation for why one and not the other, but it does seem to imply that perhaps there will be future entries in the series. I'd even say that maybe that's why they haven't continued with the Bad Company games. Battlefield 3 was well-received but has become something of a mess if you ask the fans, whilst Bad Company 2 was near perfect. Perhaps they'd rather fix their reputation with Battlefield 4 than ruin it further with Bad Company 3.