Local energy markets (LEMs) have been proposed to mitigate the variability introduced in power systems by distributed renewable energy resources such as photo-voltaic energy. During the progressive release of LEMs, the decision problem faced by prosumers (consumers that might also produce energy), will differ from the wholesale electricity market's one because there is always the alternative to buy from or sell to the utility company. In this setting, guaranteeing that the aggregated energy consumption will be well behaved depends on the properties of the mechanisms used to implement the market, the alternative tariff offered to participants by their utility and how prosumers interact among themselves. We present a pathological example of a LEM in which the best strategy for the agents results in unnecessary peaks of demand. A decision model for players participating in LEMs is developed to study the existence of undesirable behaviour while using realistic data and number of participants. Through numerical experiments, we identify the key aspects of the player's behaviour, strategy and environment that lead to the aforementioned peaks, all under reasonable circumstances. Simple fixes are discussed to overcome the pitfalls of such markets.