I am a visa officer, so forgive me for speaking bluntly:

DON'T FLASH CASH! I mean it. Even when dealing with developing nation's visa offices. You can get blacklisted, and you will have virtually no way to fix it. Just because it works once does not imply it will work always. A visa officer who is annoyed, or offended will find it far more satisfying to foresake the cash, and put you on a watch list forever. If there happens to be an audit team visiting the consulate, a visa officer might want to impress the auditors with some apparent incorrubtability.

For the 49 times it works, the 50th time will be enormously embarrassing. (Sorry boss, the Brazilians have refused my visa. Why? Umm...)

However, I sympathize. Visa regulations are arcane, and the processes are frustrating. So, from the perspective of a visa officer, here is my advice:

1) Keep your passport valid. Even if you only travel to Canada, make sure you have a valid passport. Many countries will refuse you flat out without 6 months validity. That will leave you arguing with State Department for a fast passport, and then going back to the visa office at the last minute.

2) Plan ahead. Allow at least two weeks to get a visa from most countries. If they produce it on the spot or within a few days, all to the good. If you are applying by mail, count the mailing time (at least 8 days for certified mail). Allow a month for ones whose relations with your country are strained. Sometimes there is no getting around this. Countries impose minimum processing times for many reasons, not all of which are discretionary.

If your plane ticket pre-dates your visa application, a visa officer will not be impressed by your apparent lack of planning.

3) Prepare: Make sure that you have all of
the necessary supporting documents. Letters should include as much information as possible--more detail rather than less. Visa officers typically want to know exactly where you are going, exactly what you are doing, who is paying for it, and how much. If you include too much, a visa officer can always sift through it. If you don't include enough, you get bounced to provide more detail. I can't count the number of times I have had to put off an application because an employer has sent a three-liner which tells me nothing of value, e.g. "Mr. X is travelling on business. We vouch for him. Please issue the appropriate visa."

4) Don't yell, threaten or bluster (either in person or in writing). There is nothing, and I mean nothing, that will get a visa officer's back up faster (your's truly included) than a person who argues, "But I have to be there tommorrow", "This is a stupid bureaucratic restriction" (I hate that one most of all) or demands to speak to the Consul General, the Prime Minister, etc.

5) Don't call in your congressman or senator. Non-US officials will be neither interested nor impressed. However, if you are trying to facilitate a US visa for a colleague, that is a different story.

Sorry to say it, but forget a bilateral waiver. The only country in SA that might be in the running is Chile, and Canada's experience with lifting the visa requirement on Chile a few years back was bad. Neither the US nor Canada are likely to consider lifting the visa requirement on Brazil any time soon.