No matter how often they say no. No matter how much they complain that it's not in the budget. Bring new, creative ideas to your clients - all the time! You just have to be willing to hear 'no" and hear it often. But you cannot be worn down because the day you stop bringing new ideas to your client, will be the day someone else will.
Regarding budgets, consider small budgets not as obstacles, but creative weapons. When we were little kids and wanted to give our mom or dad a birthday card, we lacked the money (or at least I did) to buy a card at the local store. Forced to seek other means, we found crayons, paper, old magazines, glue, etc. and we concocted something more special and more personal than Hallmark could ever create. It's no different servicing clients, but now you get paid for it. How great is that?
Companies hire agencies for the power of their ideas. The more ideas you bring, the more fun for you and the more value for your client.