Two radiocarbon excursions (AD 774–775 and AD 993–994) occurred due to an increase of incoming cosmic rays on a short timescale. The most plausible cause of these events is considered to be extreme solar proton events (SPE). It is possible that there are other annual 14C excursions in the past that have yet to be confirmed. In order to detect more of these events, we measured the 14C contents in bristlecone pine tree-ring samples during the periods when the rate of 14C increase in the IntCal data is large. We analyzed four periods every other year (2479–2455 BC, 4055–4031 BC, 4465–4441 BC, and 4689–4681 BC), and found no anomalous 14C excursions during these periods. This study confirms that it is important to do continuous measurements to find annual cosmic-ray events at other locations in the tree-ring record.