I'd say there is no definite answer to this question. In most instances I would say that having one card with a bank helps to get the second one, but it might be exactly the opposite - having one card basically makes you ineligible to get another one. It depends on other factors as well.
I suspect most of sophisticated banks maintain a what can be called "Internal Single Customer Exposure Limit" (might be called differently, might not exist at all etc.). This can basically be the cap on sum of all CLs they extended to you and absolutely not necessarily should be equal to your current credit limit with them.
This case if you want to get a second card with them and their risk assessment system sees that you are a good customer and your total internal "limit cap" allows adding one more card without raising it too high, then you are good and most likely approved right away.
Another outcome might be reallocation of lines - if you already have a card with a lender with huge CL and you want another card in the worst case scenario you can ask to reallocate the line (splitting it to two cards). Nonetheless nobody can guarantee that lender will do it. Maybe the most well-known illustration of this concept is Amex NPSL.
Though YMMV.
Me and my wife had negative experience with Citi and Barclays trying to obtain second cards with them. Citi ended up with sending me 4506T request in response to my second app; Barclays just outright declined my wife's app for the second card and recon call (when she asked to split her current line to two cards) was of no positive result as well.
EDITED: Just to add a little bit of a personal flavor - I currently hold 3 active Amex cards, 3 active Chase cards and 2 active BofA cards :-) So generally it is possible.
In my wallet now: Amex PRG NPSL, Amex BCE $15k $17k (thanks to recent CLI), Chase Freedom 11k, CSP $6k, Chase United ME $5k, Citi Dividend $5.6k, Discover It $4.5k and a handful of other cards...
TU Dec 2013 - 752Sep 2013 - 764 - new all time maximum
Obsolete data aboveCurrent TU - probably around ~810, only using a few Chase & BofA cards now, not applying for much until we buy a house
