The following quote comes from the BBC news website, is is grammatically correct?
[quote]
Before the attacks started the suspected gunman’s Facebook status was changed to read: "Just got out of jail, I’ve lost everything, my business, my property and to top it all off my lass has gone off with someone else.
“Watch and see what happens.”[\quote]
Yes, it’s correct. When quoting multiple paragraphs (if you’re not doing a block-quote) you use an extra opening quotation mark at the beginning of each paragraph to indicate that the quotation continues.
I have the feeling that it is somewhat of an antiquated style. Block-quotes are clearer, IMHO.
Quotation marks aren’t paired if a single quote extends past one paragraph. So if you’re quoting a long speech for example and not doing it in indented form which uses not quotation marks, they might not be paired. E.g.,
"Four score and Seven Years ago ........
…
…
… <- no ending quotation mark
"(beginning of second) paragraph
…
…" <- closing quote for end of speech.
I can’t tell if that applies to your example though. I can’t tell if it’s a single quote of two paragraphs from some place or two separate quotes.
“Quotation marks,” and yes, they do – with the exception of quoted paragraphs. When quoted material has a break in the middle, you do not close the quote, but you do open the new paragraph with a quotation mark., and close the final paragraph with another mark.
“Don’t tell me you’ve never heard that rule before,” said Tom. "I sat right next to you in third grade when Mrs. Haversack explained it.
"The mark at the beginning of each paragraph is sort of a reminder that the quote is still going on. When the quote is finished, then, and only then, do you place the closing mark.
“Do you remember now?”
I’m English, we called them speech marks when I was at school.
Thanks for the helpful replies, it just didn’t look right to me, the BBC news website is littered with spelling mistakes these days, I thought their grammar was going the same way.
Colour, centre, skilful, programme, tyre. My red pen runs out of ink when I read The Times.
They come in pairs if their timing is very, very good. 
Why? what language do you speak?
I’ll add that while all marks do come in pairs, but they can be split between paragraphs, as others have said, how you implement that is subject to whatever style guide you subscribe to, and I find that I like whatever style guide specifies that paragraph spanning quotation marks should have a space after them for clarity…
And talking of quote marks, what went wrong in the OP was that you used “” instead of “/” in your quote terminator.
The programmer in me just can’t stand unbalanced quotation marks. It’s like there’s an unmet expectation in my head holding its breath until the original quotation ends.