Is your flat mate a gambler? I'd be prepared to offer him/her a £100 bet that he/she is talking rubbish!
I've been researching answers in AB's 'Travel' section for the past 5 years. While I'm prepared to accept that fares on some low-volume, short-haul routes, operated by budget airlines, might drop shortly before travel, I've hardly ever come across medium/long-haul flights experiencing price reductions in the final month or two before departure.
The answer to your question about the best time to book is 'several months ago'!
I'd be interested to know which airline you're planning to fly with as, to the best of my knowledge, there are no direct scheduled flights from the UK to Las Américas Airport in Santo Domingo. Your best flight options (although I've not actually checked on connections) are probably with American Airlines, Air France or Delta.
Chris