In 1978, Bollobás and Eldridge [5] made the following two conjectures.

1. (C1) There exists an absolute constant $c>0$ such that, if k is a positive integer and $G_1$ and $G_2$ are graphs of order n such that $\Delta(G_1),\Delta(G_2)\leq n-k$ and $e(G_1),e(G_2)\leq ck n$, then the graphs $G_1$ and $G_2$ pack.

2. (C2) For all $0<\alpha<1/2$ and $0<c<\sqrt{1/8}$, there exists an $n_0=n_0(\alpha,c)$ such that, if $G_1$ and $G_2$ are graphs of order $n>n_0$ satisfying $e(G_1)\leq \alpha n$ and $e(G_2)\leq c\sqrt{n^3/ \alpha}$, then the graphs $G_1$ and $G_2$ pack.

Conjecture (C2) was proved by Brandt [6]. In the present paper we disprove (C1) and prove an analogue of (C2) for $1/2\leq \alpha<1$. We also give sufficient conditions for simultaneous packings of about $\sqrt{n}/4$ sparse graphs.