We study the necessity of predictive information in a class of queueing admission control problems, where a system manager is allowed to divert incoming jobs up to a fixed rate, in order to minimize the queueing delay experienced by the admitted jobs. Spencer et al. (2014) show that the system's delay performance can be significantly improved by having access to future information in the form of a lookahead window, during which the times of future arrivals and services are revealed. They prove that, while delay under an optimal online policy diverges to infinity in the heavy-traffic regime, it can stay bounded by making use of future information. However, the diversion polices of Spencer et al. (2014) require the length of the lookahead window to grow to infinity at a non-trivial rate in the heavy-traffic regime, and it remained open whether substantial performance improvement could still be achieved with less future information. We resolve this question to a large extent by establishing an asymptotically tight lower bound on how much future information is necessary to achieve superior performance, which matches the upper bound of Spencer et al. (2014) up to a constant multiplicative factor. Our result hence demonstrates that the system's heavy-traffic delay performance is highly sensitive to the amount of future information available. Our proof is based on analyzing certain excursion probabilities of the input sample paths, and exploiting a connection between a policy's diversion decisions and subsequent server idling, which may be of independent interest for related dynamic resource allocation problems.