Spanish Bankruptcy Events
The Spanish bankruptcy events are two events in Spain's scripted events file, which have nasty effects. This page is a discussion of how to deal with these events. Note that this information applies only to vanilla EU2 1.09. (1.08 also?) It does not apply to AGCEEP or any other mod.
Description of the Bankruptcy Events
There are two bankruptcy events, both of which have the same triggers and similar effects. The first event (#1) may happen from January 1, 1592, through Dec 30, 1597. The second (#2) may happen from January 1, 1621, through Dec 30, 1640. During those time windows, the event is checked once per month. The trigger is owning Aztec and/or Inca gold provinces: if you own all provinces of either country, it will trigger. In fact if you own only a subset of both, it will trigger. (The actual triggering check, and how to avoid it most easily, is discussed below.)
The effects of the bankruptcies are quite nasty. They have the following nasty effects:
Handling the Bankruptcy Events
There are basically two ways to deal with the bankruptcy events. One is take them like a man. The other is to avoid via one means or another. We discuss both below.
Take the Hits
This is by far the easiest option. Just play your game, and when the events happen, they happen. 15% inflation is nasty, but it won't kill you, and having all the gold income from the Aztecs and Incas for the maximum time is a nice thing.
You should definitely save up for the events ahead of time, so that they do not make you take costly loans. Try to avoid being in a big war when the events hit, so that the desertions don't particularly hurt. Lastly, you'll have to decide about whether to avoid building manufactories. Generally, it is economically unsound to put off investments for 100+ years just because you'll lose a few. But if your Spain is not doing great economically in the 1500s, you should consider building no manufactories until bankruptcies are over.
The bankruptcy events can be avoided. The exact triggers are as follows: they happen only if Spain owns at least one province in all three of following sets:
Jalisco (24)  Tenochtitlan (27)  Yatumal (159) 
Saltillo (25)  Azuay (163)  Ayacucho (167) 
Michoacan (26)  Cuzco (169)  Potosi (194) 
Thus, to avoid bankruptcy, you must leave one of the sets in which you own no provinces at all (during the years when the events are active). Note that all of the provinces are cities when the game starts, except for Yatumal, which is unsettled.
Generally, the easiest of the three province sets to avoid owning is the first one. Here's what you want to do. First, using Chris Columbus, find America and get at least one colony in the sugar islands. This gives you a place to stage troops, which you should do. You can ship over a good army without too much attrition.
Method 1 (No Quest)
You need perhaps 15000 men. Place 10000 of them under El Gran Capitán, and get your land tech to at least 9 for a fire phase. With his fire stat of 6, El Gran Capitán is almost unbeatable. The only way the natives can stop him is attrition. But you don't have to spend a lot of time sieging forts: assault them. Do not be afraid to tangle with Aztec armies, except for large ones which will attrit you. (The 5000 other men are reserves to replenish the main army as it gets attrited.)
Using tiny armies, discover each Aztec city in turn, the follow with the main army and assault each one. Discover Tenochtitlan as early as you can, so that you can see it. But do not capture it; leave Tenochtitlan for last. If you occupy Tenochtitlan, the Quest for the New World scripted event kicks in (within a month), in which you inherit Aztecs. You do not want that, because once you own Jalisco it is very difficult to get rid of. Usually the Aztecs stack up a huge army on Tenochtitlan. It is worth waiting to avoid tangling with it. If you wait it will eventually move to siege to retake of the other cities. That's when you move in behind it and assault Tenochtitlan. Upon success, immediately pause the game. You have 100% warscore, so you can make peace taking all provinces other than Jalisco, and as much money as you can get, but ask for no more than 89% warscore.
With the Aztecs finished, you can now turn to South America. Just conquer it as fast as you can. With limited men, you will probably want two wars: in the first, you just take one province (probably Azuay), and leave tiny armies in enemy terrain to do white flag tourism. Explore them during the truce, and blitz them in 5 years.
Now, so long as you don't settle Yatumal, you won't be hit by the bankrupties. When 1641 rolls around, you can annex the Aztecs, and settle Yatumal.
Method 2 (Uses Quest)
Alternatively, you can use the Quest for the New World event. This method is a lot more complicated and time-consuming to run. It is also on a fairly tight clock: the Aztecs cease to be a releasable/revoltable country on Jan 1, 1520. So you must arrange to conquer them and recreate them before that time.
You need maybe 10000 troops. Put them with El Gran Capitán, DoW, and beeline for Tenochtitlan. Don't stop to assault other cities. Once you capture Tenochtitlan, the Quest for the New World scripted event kicks in (within a month). Then you inherit the Aztecs. (This costs a little bit of badboy, but it is fast and cheap. Also the Quest gives you 300d, as well as 6 free colonists and 6 missionaries. So plan accordingly.)
Having conquered the Aztecs, you can release then as a vassal, and/or let Tenochtitlan revolt. I suggest sending missionaries to all nearby provinces. If they all succeed, great. Release Aztecs as a vassal, and declare war. (You can get CB on them by allying, then using insults to ruin your relationship, and then calling them to some other war.) If any adjacent missionary fails, let the rebels go. They'll capture the city they are in, then move on to the highest income adjacent city -- Tenochtitlan. (Move in behind to recapture.) The rebels will capture Tenochtitlan, and in 4 years, revolt. (This saves you the stability cost of breaking a vassal and the DoW.)
Once at war with the Aztecs, let them go. They will siege all of their core cities first, and eventually (probably taking years), get around to taking Jalisco. Once they capture Jalisco, make peace giving them whatever they want so long as Jalisco is included.
After 5 years, you need one more 100% warscore war, to go in and retake all the Aztec cities other than Jalisco.