Let ξ1, ξ2,… be a Poisson point process of density λ on (0,∞)
d
, d ≥ 1, and let ρ, ρ1, ρ2,… be i.i.d. positive random variables independent of the point process. Let C := ⋃
i≥1 {ξ
i
+ [0,ρ
i
]
d
}. If, for some t > 0, (0,∞)
d
⊆ C, then we say that (0,∞)
d
is eventually covered by C. We show that the eventual coverage of (0,∞)
d
depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of ℝ
d
by such Poisson Boolean models. In addition, we consider the set ⋃{i≥1:X
i
=1}
[i,i+ρ
i
], where X
1, X
2,… is a {0,1}-valued Markov chain and ρ1, ρ2,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.