The NSLS-II storage ring is designed to operate with superconducting RF-cavities with the aim to store an average current of 500mA distributed in 1080 bunches, with a gap in the uniform filling for ion clearing. At the early stage of the commissioning (phase 1), characterized by a bare lattice without damping wigglers and without Landau cavities, a normal conducting 7-cell PETRA-III RF-cavity structure has been installed with the goal to store an average current of 25mA. In this paper we discuss our analysis of coupled-bunch instabilities driven by the Higher Order Modes (HOMs) of the 7-cell PETRA-III RF-cavity. As a cure of the instabilities, we apply a well-known scheme based on a proper detuning of the HOMs frequencies based upon cavity temperature change, and the use of the beneficial effect of the slow head-tail damping at positive chromaticity to increase the transverse coupled-bunch instability thresholds. In addition, we discuss measurements of coupled-bunch instabilities observed during the phase 1 commissioning of the NSLS-II storage ring. In our analysis we rely, in the longitudinal case, on the theory of coupled-bunch instability for uniform fillings, while in the transverse case we complement our studies with numerical simulations with OASIS, a novel parallel particle tracking code for self-consistent simulations of collective effects driven by short and long-range wakefields.the coupled-bunch instability for uniform fillings. A justification of this assumption is based on the work of Prabhakar [13] and Kohaupt [14]. A detailed analysis of the coupled-bunch instability driven by non-uniform fillings is discussed in a separate paper [15], where a derivation of the analytical formula for the complex frequency shift induced by arbitrary fillings is given and benchmarked against numerical simulations and measurements in the 15 NSLS-II storage ring. For the instability threshold calculations based on the HOMs of the 7-cell PETRA-III cavity structure we rely on the numerical data computed by R. Wanzenberg [16], complemented by measurements aimed to determine the dependence of the HOM frequencies on the cavity temperature [17]. In the longitudinal case, since in the NSLS-II storage ring the cut-off frequency of the PETRA-III cavity is higher than the case considered by R. Wanzenberg, we computed HOMs up to 1390 MHz [18], as shown in Table 2, with the GdfidL code [19]. 20 The beam dynamics simulation results discussed in this paper have been obtained with OASIS [20], a parallel code for self-consistent simulations of single and multi-bunch effects. For single-bunch effects, the code uses the same physical model of the TRANFT code developed by M. Blaskiewicz [21], [22], while in the case of multi-bunch effects the code implements a novel self-consistent algorithm to take into account long-range effects [23]. The novelty of the algorithm consists of the capability to simulate efficiently the self-consistent coupled-bunch interaction of 25 bunches with finite length distributed in arbitrary filling patterns, v...