Nested Named Entity Recognition (NER) is an information extraction task that aims to identify entities that may be nested within other entity mentions. Despite the availability of several corpora with nested entities in the Spanish clinical domain, most previous work has overlooked them due to the lack of models and a clear annotation scheme for dealing with the task. To fill this gap, this paper provides an empirical study of straightforward methods for tackling the nested NER task on two Spanish clinical datasets, Clinical Trials, and the Chilean Waiting List. We assess the advantages and limitations of two sequence labeling approaches; one based on Multiple LSTM-CRF architectures and another on Joint labeling models. To better understand the differences between these models, we compute task-specific metrics that adequately measure the ability of models to detect nested entities and perform a fine-grained comparison across models. Our experimental results show that employing domain-specific language models trained from scratch significantly improves the performance obtained with strong domain-specific and general-domain baselines, achieving state-of-the-art results in both datasets. Specifically, we obtained F1 scores of 89.21 and 83.16 in Clinical Trials and the Chilean Waiting List, respectively. Interestingly enough, we observe that the task-specific metrics and analysis properly reflect the limitations of the models when recognizing nested entities. Finally, we perform a case study on an aggregated NER dataset created from several clinical corpora in Spanish. We highlight how entity length and the simultaneous recognition of inner and outer entities are the most critical variables for the nested NER task.