What I've done with my past two places is buy a place, pay off a chunk of it, then move elsewhere, buy another property and keep my old place as a rental. The numbers on both places worked.
I know a couple who bought 8 rentals and they are doing really well out of it as they cash-flow nicely and he is a carpenter so they can maintain them easily, but I personally wouldn't like that level of risk (I think they had 500K+ of mortgage debt). But as long as the numbers make sense, and you could service the mortgages easily if you had a long void period, or to fix up the rental if a tenant trashed it, then it makes sense.
Have you looked into the availability of financing the purchase(s)? I'm actually itching to buy another property as there are some good deals (as far as the UK goes) I am seeing, but I'm limited by the fact that without a permanent full-time job banks are hesitant to lend so the only ones on offer require a 40% deposit, which limits my buying power.