How much does a bootstrap estimate depend on the random number stream?

Many modern statistical techniques incorporate randomness: simulation, bootstrapping, random forests, and so forth. To use the technique, you need to specify a seed value, which determines pseudorandom numbers that are used in the algorithm. Consequently, the seed value also determines the results of the algorithm. In theory, if you know the seed value and the internal details of the pseudorandom algorithm, then the stream is completely determined, and the results of an algorithm are reproducible. For example, if I publish code for a simulation or bootstrap method in SAS, you can reproduce my computations as long as my program specifies the seed value for every part of the program that uses random numbers.