Predicting? Not really. You can use sophisticated quantum mechanical calculations to explore the energetics of the reaction channels, and apply let's say trainsition state theory to get dynamical informations. This method is cumbersome, time-consuming and still far from being exact - but this is as close to predicting reaction pathways as it gets.
Now, in organic chemistry, you usually go with your "gut". I will try to present a way which organic chemist can think. When you have seen a lot of mechanisms, then you can guess whether reorganization happens or not. It also helps if you enumerate the circumstances that can affect the reaction. Let's see some of these:
In order to have a rearrangement, you obviously need one part of the molecule (the "jumping" part) to have access in space to the receiving part of the molecule. This requirement can be satisfied if we let only a hydrogen atom jump, or we let the groups jump only a single bond. Now there are a lot of exceptions of this rule: for example in Benzilic acid rearrangement you can have phenyl groups mitigating. The idea that the groups have to be close to each other is also not always true, because you can have "jumps" between different molecules too. However, if the target part of the molecule is extremly hindered, then no rearrangement can be expected.
Now, if we look at the electronic requirements of a rearrangement, then we have two different things to look for. One type of reactions can be called "orbital-controlled", because the mechanism depends mostly on the frontier orbital shapes and overlap of the reactants - in this case the rearrangements. A good example of these types of reactions are the sigmatropic processes, like Cope rearrangement.
If we assume that the orbitals are alright - which is not a huge assumption if we consider hydrogen transfer - then we have to look at the free energy profile of the reaction. This means we have to check how much energy we can "win" by rearranging the reactants. In the case you mentioned with the carbocations, this is equal to checking how stable is the carbocation. If it is instable (high energy), then we can "win" a lot by rearranging the molecule to the lower energy - by mitigating a hydrogen atom let's say. The stability of the carbocation then depends on the order of the cation and the electronic effects of different substituents. The general idea is that if you do not have your charge well distributed over many atoms, then you have a high energy - for example, CH3+ cation is extremly instable. If you have a lot of groups to "help" carrying the charge, either by inductive effects, or by resonance, then the carbocation stabilizes, making the rearrangement less favoured.
It is important to note that the things I have mentioned are mostly about the energetics of the reaction. Organic chemists seldom have a way to say anything about the kinetics of a particular reaciton. There are tricks like Hammett's relations and tranisiton state theory, but most of the time, they simply are not applicable. In our case with the hydrogen transfer, we could also look for quantum effects, which is completely out of organic chemistry's predictive reach.