Let \(f\) be a nonlinear entire function for which \(F(f)\) denotes its Fatou set (i.e. the largest open set in the complex plane where the family \(\{f^n\}\) of iterates of \(f\) forms a normal family). Suppose \(f(z)\) has the power series representation \(\sum^\infty_{k=0} a_kz^{n_k}\). Then the author presents two situations in which every component of \(F(f)\) is bounded: (1) \(f\) has finite order but positive lower order and \(n_k/k \to\infty\) as \(k\to\infty\); and \[ \liminf_{r\to \infty}\bigl\{ \log M(r^T,f)/ \log M(r,f)\bigr\} >T\tag{2} \] for some number \(T>1\) and \(n_k>k\log k(\log\log k)^\alpha\) as \(k\to\infty\) for some \(\alpha>2\). The proofs, which are parallel in character and intricate in substance, make use of deep results of W. H. J. Fuchs [Ill. J. Math. 7, 661-667 (1963; Zbl 0113.28702)], W. Hayman [Proc. Lond. Math. Soc., III. Ser. 24, 590-624 (1972; Zbl 0239.30035)], and I. N. Baker [J. Aust. Math. Soc., Ser A 30, 483-495 (1981; Zbl 0474.30023)], and an argument used by Baker. Interest in the question stems from Baker’s question of whether every component of \(F(f)\) must be bounded if \(f\) is of sufficiently small growth.