In addition to satellite systems, carrier phase positioning (CPP) is gaining attraction also in terrestrial mobile networks, particularly in 5G New Radio evolution toward 6G. One key challenge is to resolve the integer ambiguity problem, as the carrier phase provides only relative position information. This work introduces and studies a multi-band CPP scenario with intra- and inter-band carrier aggregation (CA) opportunities across FR1, mmWave-FR2, and emerging 6G FR3 bands. Specifically, we derive multi-band CPP performance bounds, showcasing the superiority of multi-band CPP for high-precision localization in current and future mobile networks, while noting also practical imperfections such as clock offsets between the user equipment (UE) and the network as well as mutual clock imperfections between the network nodes. A wide collection of numerical results is provided, covering the impacts of the available carrier bandwidth, number of aggregated carriers, transmit power, and the number of network nodes or base stations. The offered results highlight that only two carriers suffice to substantially facilitate resolving the integer ambiguity problem while also largely enhancing the robustness of positioning against imperfections imposed by the network-side clocks and multi-path propagation. In addition, we also propose a two-stage practical estimator that achieves the derived bounds under all realistic bandwidth and transmit power conditions. Furthermore, we show that with an additional search-based refinement step, the proposed estimator becomes particularly suitable for narrowband Internet of Things applications operating efficiently even under narrow carrier bandwidths. Finally, both the derived bounds and the proposed estimators are extended to scenarios where the bands assigned to each base station are nonuniform or fully disjoint, enhancing the practical deployment flexibility.