We establish heavy-traffic limits for nearly deterministic queues, such as the G/D/n many-server queue. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated G n /D/1 model, where the G n interarrival times are the sum of n consecutive interarrival times in the original model, we focus on the G n /D/1 model and the generalization to G n /G n /1, where ‘cyclic thinning’ is applied to both the arrival and service processes. We establish different limits in two cases: (i) when (1 − ρ n )√n → β as n → ∞ and (ii) when (1 − ρ n )n → β as n → ∞, where ρ n is the traffic intensity in model n. The nearly deterministic feature leads to interesting nonstandard scaling.