Two affiliates have the same offer and they're both promoting it on the same network. Is it possible for it to show up at the same time?
What about if the actual company is promoting it?
For example - I see that the actual company is promoting their own offer on Bing Ads. I want to promote that same offer on Bing Ads using the same keywords. Will I be able to?
Good heads up universive! 
@deehkay: Also keep in mind that the advertiser has a better margin to play with than you do (in many cases, a MUCH better margin), so a campaign setup that's profitable for them may not be profitable for you. Therefore you'll want to avoid bidding wars with them (which you know you probably won't win!)
Perhaps you could spend more time on keyword research to see if you can find keywords that the advertiser isn't bothering to target or has neglected to target. And bidding on the same keywords as they are is worth testing as well - I don't have experience with Bing but did dabble in adwords some years ago - you may be able to bid significantly lower if you're willing to have your ads shown in a lower position.
From what I gather, many advertisers will try to maintain first ad position because they can afford to do so and because they want to maximize exposure for their ads. But I've found that top position didn't necessarily mean better conversion rates or ROI - in fact, if I remember correctly, I found the 3rd-5th spots to be the best, probably because there are much fewer tire-kickers which usually won't bother clicking on ads further down the page, and also because people that DO bother clicking on ads further down, are probably in more dire need of the particular solution/service so they often convert better. And you could be paying significantly less than the advertiser who's paying premium rates to be at the top.
So I would definitely suggest that you play with bids and observe changes in ad positions and the corresponding changes in CR/ROI, to find that sweet optimal bid and spot. 
Amy
Generally the cap is split based on the quality of your traffic.