Police said two fights broke out close to Barker's Pool in the early hours of Saturday morning leaving five people injured - four of which with stab wounds.
Four people have been arrested, including two on suspicion of attempted murder.
Members of the public and people who work in bars close by told how the incidents had put them 'on edge'.
Jessica Walton, aged 24, who lives on West Street, said: "I have only just moved here so it puts me a bit on edge.
"My family members visited and saw all the police there and were shocked.
"I'll be avoiding this area at night."
Her 24-year-old friend, who did not want to be named, added: "It always makes you wonder in this day and age if it's a terror attack, that was my first thought.
"Things like this happen in city centres, but usually it's a one-off. To have five people injured is a lot. It must have been a huge fight."
Helena Trabucchi, aged 20, who works at the Tap and Tankard in Cambridge Street, just yards from one of the incidents, said: "We couldn't get to the pub on Saturday morning because of the police cordon. It's a little scary."
The first fight was reported to police at about 4am in Cambridge Street and resulted in an Asian man suffering cuts and bruises and what police described as 'minor stab wounds. He was taken to hospital for treatment, but released and did not want to make a formal complaint.
At around 5.53am there was a second brawl in Carver Street, in which three Polish men were stabbed. All were taken to hospital and one was described by police as being in a 'serious but stable' condition.
Another man was hit over the head with a bottle during the second fight and taken to hospital to be treated for a minor head injury. He was later discharged.
Officers arrested two men close to the scene on Carver Street and two further men were arrested later in Spital Hill.
Two arrested on suspicion of attempted murder are still being questioned in custody today.
Two further men arrested for public order offences have been freed from custody. One has been released under investigation while the other faces no further action.
Superintendent Caroline Rollitt said officers are unclear over the motives of those involved at the moment and are trawling through CCTV in the area.
She said: "The first fight was a large brawl involving a number of people where an Asian male appears to have been set upon.
"We don't know if the second fight is a continuation of the first or the three Polish men were just in the wrong place at the wrong time."
She said officers "suspect there may be more people who have been injured who haven't come forward."
Supt Rollitt added: "There will have been people who probably witnesses something, other people on nights out and taxi drivers, who aren't aware of what has happened.
"We want to speak to anyone who was in the area or who witnessed anything to come forward."
Extra police were deployed to the streets on Saturday night as a precaution. Earlier in the day a cordon had been established and crime scene investigation officers in forensic gear were seen clearing away white items of clothing with what appeared to be blood on them.
Police said the attacks were not terror related following speculation on Twitter.
They also said they are unable to release any CCTV at this point as the investigation continues.
Witnesses are urged to call 101. Alternatively call Crimestoppers anonymously on 0800 555111.